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// SCAN REPORT

ScorpTech x402 API

1012 tools. 142 can modify or destroy data without limits.

11 destructive tools with no built-in limits. Policy required.

Last updated:

142 can modify or destroy data
870 read-only
1012 tools total

Community server · catalogue entry checked 30/10/2026 · full schemas captured for 1012 of 1012 tools

How to control ScorpTech x402 API ↓

TOOL MAP

What ScorpTech x402 API exposes to your agents

Read (870) Write / Execute (131) Destructive / Financial (11)

Running more than one server? Check your whole stack →

MOST DANGEROUS TOOLS
Critical Risk

The most dangerous ScorpTech x402 API tools

Destructive · Critical dedupe_records Find and remove duplicate records (CRM contacts, leads, customers, products) in a JSON array. Modes: exact, normalized (emails: case, Gmail Destructive · Critical ruler_odd Remove every factor of 2 from n. The odd part of 12 is 3 and the valuation is 2. The odd part of 7 is 7. n runs from 1 to 1000000000000. Pri Destructive · Critical sql_format Format and lint SQL: consistent keyword case, one clause per line, indented select lists, joins, AND/OR conditions and subqueries, comments Destructive · Critical timecode_duration Exact length of a frame count. 24, 25 and non-drop 30 use 1/rate second per frame. Drop-frame 30 uses the NTSC frame, 1001/30000 second. Pas

142 of ScorpTech x402 API's 1012 tools can modify, destroy, or commit something on every call — and an agent calls them with no built-in limits.

POLICY

How to control ScorpTech x402 API

PolicyLayer is an MCP gateway — it sits between your AI agents and ScorpTech x402 API, and nothing reaches the server without passing your rules. These are the rules we recommend:

Deny destructive operations
{
  "collatz_drop": {
    "deny_if": [
      {
        "conditions": [],
        "on_deny": "Blocked by default. Requires approval."
      }
    ]
  }
}

Destructive tools should never be available to autonomous agents without human approval.

Rate limit write operations
{
  "barcode_generate": {
    "limits": [
      {
        "counter": "barcode_generate_per_hour",
        "window": "hour",
        "max": 30,
        "scope": "grant"
      }
    ]
  }
}

Prevents bulk unintended modifications from agents caught in loops.

Cap read operations
{
  "abundance": {
    "limits": [
      {
        "counter": "abundance_per_minute",
        "window": "minute",
        "max": 60,
        "scope": "grant"
      }
    ]
  }
}

Controls API costs and prevents retry loops from exhausting upstream rate limits.

  1. Create a free account and register ScorpTech x402 API — nothing to install.
  2. Add these rules — paste them, or build them visually. Tune the limits to your setup.
  3. Point your MCP client (Claude, Cursor, anything) at your gateway URL.
ENFORCE POLICY ON SCORPTECH X402 API →

Instant setup, no code required.

FULL CATALOGUE

All 1012 ScorpTech x402 API tools

DESTRUCTIVE 11 tools
Destructive collatz_drop How many Collatz steps until the value is strictly below the start. 1 is already at the bottom and takes 0. 2 Destructive dedupe_records Find and remove duplicate records (CRM contacts, leads, customers, products) in a JSON array. Modes: exact, no Destructive evm_selector_decode Decode EVM function selectors, calldata and event topics: looks up 4-byte selectors (or 32-byte event topics) Destructive frames_timecode Write a frame count since midnight as SMPTE timecode. Set drop for 30 fps drop-frame. The count has to land in Destructive ip_reputation IP blocklist reputation for up to 50 IPv4/IPv6 addresses: membership in the Tor exit list, Spamhaus DROP/DROPv Destructive led_resistor Series resistor for LEDs on one supply. Resistance is (supply − series × forward) / current, with volts and am Destructive ruler_odd Remove every factor of 2 from n. The odd part of 12 is 3 and the valuation is 2. The odd part of 7 is 7. n run Destructive sql_format Format and lint SQL: consistent keyword case, one clause per line, indented select lists, joins, AND/OR condit Destructive timecode_diff How many frames from one timecode to another, at the same rate. Positive means to is later. Both codes use a c Destructive timecode_duration Exact length of a frame count. 24, 25 and non-drop 30 use 1/rate second per frame. Drop-frame 30 uses the NTSC Destructive timecode_frames Count the frames in an SMPTE timecode from midnight. Rates are 24, 25 and 30. A colon is non-drop and a semico
EXECUTE 73 tools
Execute abi_encode Build EVM calldata offline: function selector + ABI-encoded arguments from a human signature such as transfer( Execute apery_term One summand of the Apery number, C(n,k)^2 C(n+k,k)^2. The k=1 summand of A(2) is 36 and the k=0 summand is 1. Execute ballot The ballot number B(n, k) = (n-k+1)/(n+1) binom(n+k, k), for 0 <= k <= n. B(n, n) is the nth Catalan number. Execute baum_sweet The Baum-Sweet bit of n. It is 1 when every run of 0s in the binary expansion has even length, and 0 when some Execute binom The binomial coefficient C(n, k). k must not exceed n, and both indexes run up to 4096. An optional modulus re Execute bit_subset Whether every 1 bit of a is also a 1 bit of b. 3 is a bit-subset of 7. 3 is not a bit-subset of 5. 0 is a bit- Execute biunitary_sigma The sum of the bi-unitary divisors of n. For a prime power p^e the exponents run from 0 through e, skipping th Execute champernowne_start The first Champernowne place occupied by the integer m. 1 starts at 1, 10 starts at 10, 11 starts at 12, and 1 Execute clinical_trials Search ClinicalTrials.gov (API v2) by condition, keyword, location, sponsor, recruitment status (or open/close Execute coin_min The fewest coins of denominations a and b that sum to n. Making 10 with 2 and 5 takes 2 coins. An amount that Execute coin_ways The number of non-negative integer solutions of ax + by = n. Order of summands does not matter. There are 2 wa Execute color_palette Generate colour palettes from one base colour: complementary, analogous, triadic, split-complementary, tetradi Execute compound_interest Compound growth / savings calculator: future value of a principal plus monthly contributions at any compoundin Execute cron_explain Explain a standard 5-field cron expression in plain English and list its next N run times (default 5, max 20) Execute curl_to_code Convert a curl command (copied from browser dev tools, API docs or a terminal) into ready-to-run code: Python Execute date_calc Calendar-correct date arithmetic in any IANA zone: add or subtract durations (ISO 8601 or plain English, month Execute delannoy The Delannoy number D(m,n), the number of paths from (0,0) to (m,n) using steps (1,0), (0,1), and (1,1). D(3,5 Execute delannoy_sum The sum of Delannoy numbers D(m,0) through D(m,hi). The sum through D(4,4) is 501. m and hi run from 0 to 64. Execute dmarc_report_parse Parse DMARC aggregate (RUA) reports and SMTP TLS reports (TLS-RPT, RFC 8460) into an actionable summary. Accep Execute dockerfile_lint Offline Dockerfile linter: parses stages/continuations and runs 20+ best-practice and security rules (latest t Execute domb_term One summand of the Domb number, C(n,k)^2 C(2k,k) C(2n-2k,n-k). The k=1 summand of D(2) is 16 and the k=0 sum Execute duration_parse Parse any duration into seconds and every common format: ISO 8601 (P1DT2H30M, PT1.5S, P2W), clock (1:02:03.5, Execute earthquakes Recent earthquakes from the USGS: filter by bounding box or center point + radius, minimum magnitude and time Execute egyptian The greedy Egyptian-fraction denominators of a/b, written comma-separated. 3/7 gives 3,11,231. 5/6 gives 2,3. Execute egyptian_first The first denominator in the greedy Egyptian expansion of a/b. 3/7 starts at 3, 5/6 starts at 2, and 2/3 start Execute egyptian_last The last denominator in the greedy Egyptian expansion of a/b. 3/7 ends at 231, 8/11 ends at 4070, and 1/2 ends Execute egyptian_length How many unit fractions the greedy Egyptian expansion of a/b uses. 1/2 uses 1, 5/6 uses 2, 3/7 uses 3, and 8/1 Execute egyptian_sum The sum of the denominators in the greedy Egyptian expansion of a/b. 3/7 sums to 245, 5/6 sums to 5, and 8/11 Execute email_parse Parse a raw email (.eml / RFC 5322 + MIME) for triage: decoded headers (RFC 2047), addresses, Received hop cha Execute entringer The Entringer number E(n, k). E(1, 1) is 1, and E(n, k) = E(n, k-1) + E(n-1, n-k). E(5, 2) is 10, and E(6, 3) Execute epss_score FIRST EPSS exploit-prediction scores for up to 100 CVEs: probability of exploitation in the next 30 days, perc Execute file_mode Explain a POSIX file mode. Accepts octal (755, 0755, 4755) or the 9-character ls form, with an optional leadin Execute geo_distance Geodesic distance and bearings on the WGS-84 ellipsoid: Vincenty (millimetre accuracy) plus great-circle haver Execute gha_workflow_lint Security audit of a GitHub Actions workflow YAML (zizmor / actionlint style): pwn-request checkouts under pull Execute hamming_dist The Hamming distance of a and b: how many bit positions differ. 13 and 7 differ in 2 places. 0 and 0 differ in Execute hockey_stick The hockey-stick sum of binomial coefficients C(i, r) from i = r through n. It equals C(n+1, r+1). The sum for Execute ics_parse Parse an iCalendar (.ics) feed from a URL (SSRF-safe; Google / Outlook / Apple calendar exports) or inline tex Execute josephus The surviving seat in the Josephus circle. Seats are 1 through n, counting starts at 1, and every k-th remaini Execute kolakoski_run The Kolakoski run that contains the 1-based index n. A run has length 1 or 2. Index 1 is a single 1. Index 2 s Execute kronecker The Kronecker symbol (a/n) for any integers a and n. (2/7) is 1, (3/7) is -1, (2/4) is 0, and (1/0) is 1. Both Execute kummer The number of carries when a and b are added in prime base p. By Kummer, that is the exponent of p in C(a+b, a Execute lah The unsigned Lah number L(n, k). L(0, 0) is 1, L(n, 0) is 0 for n > 0, and L(n, 1) is n!. L(5, 2) is 240, and Execute leyland The Leyland number x^y + y^x. 2 and 2 give 8, 2 and 3 give 17, and 3 and 3 give 54. x and y each run from 2 to Execute look_say The nth look-and-say term, starting from 1. Term 1 is 1, term 4 is 1211, and term 5 is 111221. Each run of k c Execute lucky_twin Whether n and n+2 are both lucky. 1, 7, and 13 start a lucky pair. 3 does not, because 5 is not lucky. n runs Execute markdown_table Build or parse Markdown (GitHub-flavoured) tables. Build: from an array of objects, an array of arrays or CSV, Execute mode_diff Bits that differ between two POSIX modes. added and removed run from the special bits down through owner, grou Execute morse_run The Thue-Morse run that contains n. The bit is the parity of the 1-bits of n. A run has length 1 or 2. Index 0 Execute multichoose The number of ways to choose k elements from n types with repetition, which is C(n+k-1, k). Choosing 2 from 3 Execute music_scale Spell a scale on letters from a tonic. Major, minor, harmonic minor, melodic minor (the ascending form), the c Execute nim_sum The nim-sum of a and b, their bitwise exclusive or. 13 xor 7 is 10, and 255 xor 256 is 511. Both run from 0 to Execute oauth_pkce OAuth 2.0 PKCE helper (RFC 7636 / OAuth 2.1): generate a code_verifier with its S256 code_challenge plus rando Execute paper_scale The largest scale that places one sheet on another, turning it when that lets it be larger. The scale is an ex Execute powersmooth Whether n is y-power-smooth: every maximal prime-power factor p^k is at most y. 12 is 4-power-smooth and not 3 Execute prorate Prorate a money amount across a date range, in exact cents. The window is half-open: from the start date up to Execute prufer_decode The tree edges for a Prüfer code, as a flat list of sorted pairs. Code 2,3 is the path 1-2-3-4. Code 1,1 is th Execute prufer_encode The Prüfer code of a tree on vertices 1..n. Edges are a flat list of pairs. The path 1-2-3-4 gives 2,3. The st Execute pythag_gen The Pythagorean triple from m > n >= 1 and scale k >= 1. a is k(m^2-n^2), b is k(2mn), and c is k(m^2+n^2). pr Execute random_draw Provably fair random draws for giveaways, raffles, games and A/B assignment. Pick winners from a list (unique Execute regression_fit Fit a regression model and get the full statistical table, like R's summary(lm) or statsmodels. Linear, polyno Execute resistor_code IEC 60062 resistor color code. Decode 3 to 6 bands, or encode a resistance that lands exactly on a digit-and-m Execute rough Whether n is y-rough: its smallest prime factor is at least y. 35 is 5-rough and not 7-rough. 1 is 1-rough and Execute smooth Whether n is y-smooth: its largest prime factor is at most y. 12 is 3-smooth and not 2-smooth. 1 is 1-smooth. Execute stirling2 The Stirling number of the second kind S(n,k), the number of ways to partition n objects into k nonempty unlab Execute stirling2_rec The Stirling recurrence S(n,k) = kS(n-1,k) + S(n-1,k-1). term is the first summand and diag is the second. n a Execute text_chunk Split text into retrieval-ready chunks for RAG and embeddings: recursive (paragraph > line > sentence > clause Execute text_fit Largest Helvetica size, on a fixed step, at which one line still fits a width. Sizes run from minSize to maxSi Execute url_parse Parse, normalize and clean URLs offline: scheme, credentials (redacted), host type, port, path segments, decod Execute usaspending_awards Federal contracts, IDVs, grants, loans and direct payments from USAspending.gov by recipient, awarding agency Execute user_agent_parse Parse User-Agent strings: browser and version, rendering engine, OS and version, device type (mobile, tablet, Execute vandermonde The Vandermonde convolution. The sum of C(m, k)C(n, r-k) equals C(m+n, r). For m = 3, n = 3, and r = 2 the val Execute xml_xpath Run XPath 1.0 queries over XML you supply (RSS/Atom, SOAP, sitemaps, SVG, config) or over HTML for scraping: r Execute zeckendorf The Zeckendorf representation of n. Fibonacci numbers start F1 = 1, F2 = 1, F3 = 2, and no two used indices ar
WRITE 58 tools
Write barcode_generate Generate a 1D barcode as SVG (plus base64 data URI and the raw module pattern): Code 128 (any ASCII, automatic Write base_convert Convert an integer from one base to another. Bases run from 2 to 36, and letters may be any case. A digit that Write bell The nth Bell number, the number of partitions of a set of n objects. B(0) and B(1) are 1, and B(5) is 52. n ru Write black_scholes European option pricing (Black-Scholes-Merton with continuous dividend yield): call and put prices, delta, gam Write champernowne_len How many digits are used to write the integers from 1 through m. Through 9 there are 9, through 10 there are 1 Write chmod_apply Apply a chmod mode to a starting mode. The grammar is GNU chmod: symbolic [ugoa]([-+=][rwxXst] or one of ugo), Write color_convert Convert a colour between HEX, RGB(A), HSL, HSV, CMYK, CIE Lab and OKLCH (any CSS colour syntax or one of the 1 Write coord_convert Convert a location between coordinate formats in one call: decimal degrees, DMS, degrees-decimal-minutes, UTM Write csv_to_sql Turn CSV/TSV into SQL: CREATE TABLE with inferred column types (integer/bigint, numeric, boolean, date, timest Write cube_min The fewest positive cubes that add to n. 0 takes 0, 1 takes 1, 8 takes 1, 9 takes 2, 23 takes 9, and 29 takes Write dudeney Whether n is a perfect cube and its digits add to the cube root. 1, 512, 4913, 5832, 17576, and 19683 are. 100 Write ens_resolve Resolve an ENS name to its Ethereum address (plus optional text records like avatar, url, com.twitter), or rev Write evm_create_address Predict EVM contract addresses before deployment: CREATE (deployer + nonce, RLP-encoded; up to 50 consecutive Write federal_register Search the U.S. Federal Register (federalregister.gov) for rules, proposed rules, notices, presidential docume Write file_type_detect Identify what a file really is from its bytes (not its name): 80+ signatures (images, audio, video, archives, Write float_round Round an exact decimal or fraction to binary32 or binary64. Modes are nearestEven, towardZero, towardPositive, Write format_convert Convert data between CSV, TSV, JSON, YAML and XML in one call (auto-detects the input format). CSV header hand Write fubini The nth Fubini number, the ordered partitions of an n-element set. a(0) is 1, a(2) is 3, a(3) is 13, and a(5) Write holidays_business_days Public holidays for 100+ countries (optionally one state/region) and business-day math in one call: list holid Write html_select Scrape a public page (SSRF-safe fetch) or an HTML string with a CSS selector (CSS3 plus :has, :contains, :nth- Write ics_generate Generate a standards-compliant iCalendar (.ics, RFC 5545) file from JSON events — ready for Google Calendar, O Write id_generate Generate unique IDs: UUID v4 (random), v7 (time-ordered, monotonic within a batch), v5 (deterministic from nam Write image_resize Resize / convert an image (GD): width/height up to 4096, fit contain|cover|fill|inside, output png|jpeg|webp|g Write ip_range_merge Aggregate IP allow/deny lists: merges any mix of IPv4/IPv6 addresses, CIDRs, "a-b" ranges and "ip netmask" ent Write jacobsthal_add The Jacobsthal addition rule. (L(a)J(b) + J(a)L(b)) / 2 is J(a+b). a+b must be at most 8192. Price: $0.001 USD Write josephus_k2 The Josephus survivor when every 2nd person is eliminated. Write n = 2^m + l with l < 2^m. The survivor is 2l Write json_patch Apply a JSON Patch (RFC 6902: add, remove, replace, move, copy, test) or a JSON Merge Patch (RFC 7386) to a JS Write jwk_convert Convert public keys between PEM, JWK and OpenSSH formats and compute their identifiers — what an agent needs t Write jwt_sign Create or verify HMAC-signed JWTs (HS256 / HS384 / HS512) — for test fixtures, service-to-service tokens, webh Write kaprekar Whether n is a Kaprekar number. Split n^2 so the right part has as many digits as n; the two parts add to n, a Write lychrel Whether 50 reverse-and-add steps fail to reach a palindrome. 196 and 295 do. 89, 121, and 10 do not. n runs fr Write mac_local A stable locally administered unicast MAC from a seed. SHA-256 of the seed, first six bytes, multicast bit cle Write markup_margin Price, margin and markup in exact cents. Analyze a cost and a price (margin is profit / price, markup is profi Write mcnugget_min The fewest boxes of 6, 9, and 20 that add to n. 0 takes 0, 6 takes 1, 12 takes 2, and 44 takes 4. An amount th Write mcnugget_ways How many combinations of 6, 9, and 20 boxes add to n. 0 has 1 empty order, 6 has 1, 18 has 2, 43 has 0, and 44 Write meeting_planner Find the best meeting times across time zones: give 2-20 IANA zones (optionally with per-person hours like Asi Write package_risk npm or PyPI package risk/typosquat check: registry metadata, weekly downloads (npm), deprecation, license/repo Write palindrome_reach The palindrome reached by reverse-and-add. 10 reaches 11, 19 reaches 121, and 89 reaches 8813200023188. A pali Write palindrome_steps How many reverse-and-add steps turn n into a palindrome. 7 and 121 take 0, 10 takes 1, 19 takes 2, and 89 take Write paper_size ISO 216 A, B and C sizes, plus the DL envelope, in millimeters. Portrait, so the width is the shorter side. In Write paperfold The regular paperfolding bit of n. Write n = m 2^k with m odd. The bit is 1 when m is 1 mod 4, and 0 when m i Write password_generate Generate strong secrets with a CSPRNG (never stored or logged): random passwords (length, character classes, c Write poly_add Add two integer polynomials, lowest degree first. Trailing zeros are dropped, and the zero polynomial is an em Write profanity_filter Profanity, sexual-language and slur filter for chat, usernames, reviews and LLM output: token-level matching ( Write qr_generate Generate a QR code (PNG or SVG, base64 + data URI) for any UTF-8 text up to 2953 bytes. Options: ecc L|M|Q|H, Write reverse_add n plus its digit reversal. 10 gives 11, 19 gives 110, and 121 gives 242. A trailing zero reverses as a leading Write selector_specificity Specificity of one CSS selector, from Selectors Level 4. a is IDs. b is classes, attributes and pseudo-classes Write sol_validators Solana validator set from getVoteAccounts: active and delinquent counts, total active stake, Nakamoto coeffici Write square_obstruction Write n = 4^power odd, with odd not divisible by 4. blocked is true when odd is congruent to 7 mod 8, which i Write text_diff Diff two texts: unified line diff byte-compatible with GNU diff -u (hunks, context, "No newline at end of file Write text_similarity Document similarity and near-duplicate detection: compare two texts (a, b) or cluster up to 200 texts. Metrics Write timecode_add Add a signed frame count to a timecode and stay inside the same day. A result before midnight or at the next m Write timestamp_convert Convert any timestamp: auto-detects Unix seconds / milliseconds / microseconds / nanoseconds, ISO 8601 / RFC 3 Write timezone_convert Convert a time between IANA time zones with DST awareness: local time, UTC offset, abbreviation and DST flag o Write totp TOTP / HOTP two-factor codes (RFC 6238 / 4226), compatible with Google Authenticator, Authy, 1Password and pyo Write unit_convert Unit conversion across 14 categories with exact defined factors: length, mass, volume (US and imperial), area, Write wage_calc Gross pay in exact cents. Hourly: regular hours at the rate, overtime hours at a multiplier (default 1.5) and Write zumkeller_subset One set of divisors that sums to half the divisor-sum, comma-separated, taking the earliest divisors. 6 gives
READ 870 tools
Read abundance sigma(n) minus 2n. A perfect number gives 0, an abundant number gives a positive value, and a deficient number Read abundancy The abundancy sigma(n)/n as a reduced fraction. 1 is 1/1, 4 is 7/4, 6 is 2/1, and 12 is 7/3. n runs from 1 to Read achilles Whether n is an Achilles number: powerful, but not a perfect power. 72 and 108 are Achilles. 36 is a square, a Read additive_persistence Additive persistence: how many times the decimal digits of n must be summed to reach a single digit, and that Read airport_status US airport delay status from the FAA NAS Status feed (ground delay programs, ground stops, closures, arrival/d Read aliquot The aliquot part of n, the sum of proper divisors. The aliquot part of 1 is 0, of 9 is 4, and of 6 is 6. n run Read almost_perfect Whether sigma(n) = 2n - 1. Powers of two, including 1, are almost perfect. 16 gives sigma 31. 6 and 10 are not Read alt_factorial The alternating factorial af(n) = n! - (n-1)! + (n-2)! - ... ending at 1. af(1) is 1, af(3) is 5, and af(4) is Read alt_sum The sum of alternating factorials from lo through hi inclusive. The sum from af(1) through af(5) is 127. lo mu Read alternating_factorial The alternating factorial a(n) = n! - a(n-1), with a(1) = 1. a(3) is 5, a(4) is 19, a(5) is 101, a(6) is 619, Read alternating_mod The alternating factorial a(n) modulo m. a(5) mod 10 is 1, a(6) mod 100 is 19, and a(4) mod 5 is 4. n runs fro Read alternating_next The smallest m at least n, and at most 12, whose alternating factorial is prime. 1 gives 3, 3 stays 3, and 9 g Read alternating_prime Whether the alternating factorial a(n) is prime. It is for 3, 4, 5, 6, 7, 8, and 10. It is not for 1, 2, 9, or Read alternating_sum The sum of the alternating factorials from 1 through n. Through 1 it is 1, through 3 it is 7, through 5 it is Read amicable Whether n is amicable, and its partner when it is. The partner of 220 is 284, and the partner of 284 is 220. A Read apery The Apery number A(n), the sum of C(n,k)^2 C(n+k,k)^2 for k from 0 to n. A(0) is 1, A(1) is 5, A(2) is 73, A( Read apery_mod A(n) modulo m. A(1) mod 2 is 1, A(5) mod 10 is 5, and A(8) mod 1000 is 425. n runs from 0 to 11 and m runs fro Read apery_prime Whether the Apery number A(n) is prime. A(1) and A(2) are prime. A(0), A(3), A(5), and A(8) are not. n runs fr Read apery_sum The sum of the Apery numbers A(0) through A(n). Through 0 it is 1, through 2 it is 79, and through 5 it is 853 Read arithmetic_count How many arithmetic numbers are at most n. Through 1 there is 1, through 2 there is 1, through 6 there are 4, Read arithmetic_derivative The arithmetic derivative of n. 0 and 1 give 0, a prime gives 1, 4 gives 4, 6 gives 5, 8 gives 12, and 12 give Read arithmetic_mean The average of the divisors of n, when that average is an integer. 1 gives 1, 3 gives 2, 6 gives 3, 7 gives 4, Read arithmetic_next The smallest arithmetic number that is at least n. 1 stays 1, 2 gives 3, 4 gives 5, 8 gives 11, and 13 stays 1 Read arithmetic_number Whether the average of the divisors of n is an integer. 1, 3, 6, 15, and 20 are. 2, 4, 8, and 12 are not. n ru Read arithmetic_second The second arithmetic derivative of n. 6 goes 5 then 1. 8 goes 12 then 16. 4 stays 4. 0 and 1 stay 0. n runs f Read arithmetic_sum The sum of the arithmetic numbers that are at most n. Through 1 it is 1, through 6 it is 15, through 15 it is Read arxiv_search Search arXiv papers: title, authors, full abstract, categories, dates, DOI/journal ref and direct abs + PDF li Read automorphic Whether n^2 ends with n. 1, 5, 6, 25, 76, 376, 625, and 9376 do. 10 does not. n runs from 1 to 10000. Price: $ Read balanced_prime Whether n is prime and equal to the average of the previous prime and the next prime. 5, 53, 157, and 1103 are Read balanced_ternary The balanced ternary digits of n. -1 is written as T. 0 is "0", 2 is "1T", 4 is "11", 5 is "1TT", and 6 is "1T Read bank_id_validate Validate bank routing identifiers offline (IBAN is covered by /v1/iban-validate): US ABA routing numbers (3-7- Read beatty The Beatty term floor(nsqrt(d)). floor(5sqrt(2)) is 7, and floor(2sqrt(3)) is 3. n runs from 0 to 1000000 and Read beatty_sqrt2 floor(nsqrt(2)). n=1 gives 1, n=2 gives 2, n=3 gives 4, n=5 gives 7, and n=10 gives 14. n runs from 1 to 10000 Read beatty_sqrt2_comp The Beatty complement of sqrt(2), floor(n(2+sqrt(2))). n=1 gives 3, n=2 gives 6, n=3 gives 10, and n=5 gives 1 Read bell_sum The sum of Bell numbers from lo through hi inclusive. lo must not pass hi. Each index runs from 0 to 80. Past Read bernoulli The nth Bernoulli number as a reduced fraction, from the Akiyama-Tanigawa algorithm. B0 is 1, B1 is 1/2, B2 is Read bernoulli_denom The von Staudt denominator of B(m) for even m at least 2: the product of primes p such that p-1 divides m. The Read bernoulli_sum The sum of Bernoulli numbers B0 through Bn, as a reduced fraction. The sum through B1 is 3/2, and through B2 i Read betrothed Whether n and m are betrothed: the sum of proper divisors of n is m+1, and the sum for m is n+1, with m differ Read bezier_cubic Point on a cubic Bezier at t, as an exact fraction. t is between 0 and 1. The arch (0,0), (0,1), (1,1), (1,0) Read bezier_derivative Control points of the derivative of a quadratic or cubic Bezier. A cubic derivative is a quadratic, and each c Read bezier_quadratic Point on a quadratic Bezier at t, as an exact fraction. (0,0), (1,1), (2,0) at one half is (1, 1/2). Price: $0 Read bezier_split Split a quadratic or cubic Bezier at t by de Casteljau. Left and right are the two new control polygons, and t Read bezier_tangent Tangent vector of a quadratic or cubic Bezier at t. It is the derivative curve evaluated at t. The arch (0,0), Read big_omega The number of prime factors of n counted with multiplicity. Omega(1) is 0, Omega(4) is 2, and Omega(12) is 3. Read binary_partition The number of binary partitions of n, partitions into powers of two. b(0) is 1. Odd n copies b(n-1). Even n is Read binary_partition_sum The sum of the binary partition numbers from b(0) through b(n). Through 0 it is 1, through 4 it is 10, through Read bit_length The bit length of n: 0 for 0, and one more than the position of the highest 1 bit otherwise. 1 has length 1, 2 Read bit_reverse Reverse the lowest width bits of n. 13 reversed in width 4 is 11. 1 reversed in width 8 is 128. n must be smal Read biunitary_abundant Whether the bi-unitary divisors of n sum to more than 2n. 24 and 48 do. 6, 16, and 36 do not. n runs from 1 to Read biunitary_count How many bi-unitary divisors n has. 1 has 1, 4 and 9 have 2, 8 has 4, and 16 has 4. n runs from 1 to 1000000. Read biunitary_deficient Whether the bi-unitary divisors of n sum to less than 2n. 1, 16, and 36 do. 6, 24, and 48 do not. n runs from Read biunitary_perfect Whether the bi-unitary divisors of n sum to 2n. 6, 60, and 90 do. 16, 24, 36, and 48 do not. n runs from 1 to Read bls_series Any BLS time series by id (CPI, payrolls, unemployment, state/metro LAUS, PPI, JOLTS, ECI ...) for a year rang Read blum Whether n is a product of two distinct primes that are both 3 mod 4. 21, 33, and 77 are. 9, 15, 25, 35, and 49 Read blum_count How many Blum integers are at most n. Through 20 there are 0, through 21 there is 1, through 33 there are 2, a Read blum_factors The two prime factors of a Blum integer, smaller first. 21 gives 3,7. 33 gives 3,11. 77 gives 7,11. A non-Blum Read blum_next The smallest Blum integer that is at least n. 1 gives 21, 22 gives 33, 34 gives 57, and 78 gives 93. n runs fr Read blum_sum The sum of the Blum integers that are at most n. Through 21 it is 21, through 33 it is 54, through 77 it is 25 Read bond_calc Fixed-coupon bond math: price from yield or yield to maturity from price (clean and dirty, with accrued intere Read brocard Whether n! + 1 is a perfect square. 4! + 1 = 5^2, 5! + 1 = 11^2, and 7! + 1 = 71^2. The root is 0 when it is n Read byte_sum The two historic Unix sums, matching GNU sum. bsd rotates the 16-bit sum right and counts 1024-byte blocks. sy Read cake_number The cake number, the most pieces of space cut by n planes: (n^3 + 5n + 6)/6. Zero planes give 1, three give 8, Read calkin_wilf The Calkin-Wilf tree. Give n to get the nth positive rational, starting at 1/1, or give a reduced num and den Read cantor_bit The Cantor bit of n. It is 1 when every base-3 digit of n is 0 or 1, and 0 when a digit is 2. Bit 0 is 1, bit Read cantor_count How many integers from 0 through n have only the digits 0 and 1 in base 3. The count through 0 is 1, through 2 Read card_validate Payment card number checks without storing anything: Luhn checksum, brand / network from the IIN (Visa, Master Read carmichael Carmichael's lambda of n, the exponent of the multiplicative group mod n. lambda(1) is 1, lambda(8) is 2, lamb Read carmichael_count How many Carmichael numbers are at most n. Through 560 there are 0, through 561 there is 1, through 1729 there Read carmichael_factors The prime factors of a Carmichael number, ascending and comma-separated. 561 gives 3,11,17. 1105 gives 5,13,17 Read carmichael_next The smallest Carmichael number that is at least n. 1 gives 561, 562 gives 1105, 2000 gives 2465, and 2822 give Read carmichael_number Whether n is a Carmichael number: composite, squarefree, and p-1 divides n-1 for every prime p dividing n. 561 Read carmichael_sum The sum of the Carmichael numbers that are at most n. Through 561 it is 561, through 1105 it is 1666, through Read carol The nth Carol number, (2^n - 1)^2 - 2. n=1 gives -1, n=2 gives 7, and n=3 gives 47. n runs from 1 to 128. Past Read cas_validate CAS Registry Number check digit. Two to seven digits, a hyphen, two digits, a hyphen and a check digit. No lea Read catalan The nth Catalan number. C0 and C1 are 1, C2 is 2, C3 is 5, and C_n = binom(2n, n) / (n+1). n runs from 0 to 20 Read cayley The number of trees on n labeled vertices, n^(n-2). One vertex gives 1, two give 1, three give 3, and four giv Read cayley_rooted The number of rooted trees on n labeled vertices, n^(n-1). Four vertices give 64. n runs from 1 to 30. Past 15 Read census_geographies US Census Bureau geocoder: a one-line US street address (or a lat/lon) to matched address and coordinates plus Read centered_polygonal The nth centered s-gonal number, sn(n-1)/2 + 1. Centered triangular 4 is 19, centered square 2 is 5, and cente Read central_binomial The nth central binomial coefficient (2n)!/(n!)^2. C(0) is 1, C(1) is 2, C(5) is 252, and C(10) is 184756. n r Read central_index The index n with central binomial coefficient C(n) equal to value. 1 is index 0, and 2 is index 1. A number th Read central_mod C(n) mod m for the central binomial coefficients. C(10) mod 7 is 5. The modulus is a positive decimal integer, Read central_ratio The ratio C(n)/C(n-1) = (4n-2)/n in lowest terms. For 1 it is 2/1, for 2 it is 3/1, and for 3 it is 10/3. n ru Read central_sum The sum of central binomial coefficients from lo through hi. The sum from C(0) through C(5) is 351. lo must no Read cf_best Best rational approximation with a denominator up to maxDen. It searches the convergents and the intermediate Read cf_convergent One convergent of a continued fraction. Index 0 is a0/1. The last index is the value itself. 13/8 at index 3 i Read cf_expand Continued fraction of a non-negative rational, by the Euclidean algorithm. 13/8 is [1; 1, 1, 1, 2]. 355/113 is Read cf_sqrt Continued fraction of a square root. sqrt(2) is [1; 2, 2, 2, ...] and the period is 1. sqrt(3) has period 2. A Read cf_value Exact value of a continued fraction [a0; a1, a2, ...]. Later terms are at least 1. [3; 7, 15, 1] and [3; 7, 16 Read champernowne_digit The digit in place pos of 123456789101112..., counting from 1. Place 9 is 9, place 10 is 1, place 11 is 0, and Read champernowne_owner Which integer owns place pos in the Champernowne digit stream. Place 10 and 11 belong to 10, place 12 belongs Read champernowne_place Where place pos sits inside its integer, counting from 1 on the left. Place 10 is position 1 of 10, place 11 i Read chart_svg Render a chart to a standalone SVG (plus base64 data URI) for reports, dashboards, emails and agent answers: b Read chebyshev_disc The Chebyshev discriminant identity T(n,x)^2 - (x^2-1)U(n-1,x)^2 = 1. gap is that difference. n runs from 1 to Read chebyshev_prod The product identity 2T(a,x)T(b,x) = T(a+b,x) + T(|a-b|,x). twice is the left side and sum is the right side. Read chebyshev_rec One step of the Chebyshev recurrence. kind is t or u. prev is degree n-1, older is degree n-2, and value is 2x Read chebyshev_t The Chebyshev polynomial of the first kind T(n,x) at an integer x. T(0,x) is 1, T(1,x) is x, and T(2,3) is 17. Read chebyshev_u The Chebyshev polynomial of the second kind U(n,x) at an integer x. U(0,x) is 1, U(1,x) is 2x, and U(3,2) is 5 Read chen_prime Whether n is prime and n+2 is either prime or a product of two primes, counted with multiplicity. 2, 41, and 4 Read cidr_calc IPv4 / IPv6 subnet calculator: network and broadcast/last address, netmask, wildcard, first/last usable host, Read cipher_suite_info Look up TLS cipher suites by IANA name, OpenSSL name, GnuTLS name or hex code (0x13,0x01 / 0xC02F) — one or a Read cisa_kev CISA Known Exploited Vulnerabilities check: is a CVE (or up to 100) in CISA's KEV catalog, with date added, fe Read coin_check Whether n can be made with non-negative counts of coins a and b. 10 can be made with 2 and 5. 3 cannot. 0 can. Read coin_count How many positive integers cannot be made with coprime denominations a and b. The count is (a-1)(b-1)/2. For 2 Read collatz_glide The Collatz glide of n, the number of steps until a value strictly below n. The glide of 1 is 0, the glide of Read collatz_length The number of Collatz steps from n until 1. The length of 1 is 0, the length of 6 is 8, and the length of 27 i Read collatz_max The largest value on the Collatz path from n through 1, including n. 1 peaks at 1, 3 peaks at 16, 7 peaks at 5 Read collatz_next One Collatz step. An even number is halved and an odd number goes to 3n+1. 1 goes to 4, 2 goes to 1, 3 goes to Read collatz_odd How many odd terms appear before the Collatz path reaches 1. The final 1 is not counted. 1 and 2 give 0, 3 and Read collatz_parity The parity of each term on the Collatz trajectory from n through 1. 0 is even and 1 is odd. The parity of 6 is Read collatz_peak The largest value on the Collatz trajectory from n through 1, including n. The peak of 1 is 1, the peak of 6 i Read collatz_step One Collatz step. An even n is halved and an odd n becomes 3n+1. 6 goes to 3, and 1 goes to 4. n runs from 1 t Read collatz_steps How many Collatz steps it takes to reach 1. 1 takes 0, 2 takes 1, 3 takes 7, 6 takes 8, 7 takes 16, and 27 tak Read company_profile Company profile from its own website: name, description, logo, social profiles, contact emails/phones, postal Read complexity The integer complexity of n, the fewest 1s added and multiplied to reach n. 1 is 1, 6 is 5, 12 is 7, and 23 is Read complexity_count How many integers from 1 through n have the same complexity as n. 5 has 1, 6 has 2, 10 has 1, and 12 has 2. n Read complexity_least The smallest positive integer whose complexity is k. k=1 gives 1, k=6 gives 7, k=8 gives 11, k=11 gives 23, an Read complexity_max The largest integer complexity attained from 1 through n. Through 1 it is 1, through 10 it is 7, and through 1 Read complexity_sum The sum of the integer complexities from 1 through n. Through 1 it is 1, through 10 it is 45, and through 12 i Read contact_extract Extract public business contact details from a website: email addresses (mailto, visible text incl. [at]/[dot] Read container_check ISO 6346 container check digit. Four letters (the fourth is U, J, Z or R), six serial digits and a check digit Read cosine_law The cosine of the angle opposite side c, as a reduced rational (a^2+b^2-c^2)/(2ab). The sides must form a stri Read cototient The cototient n - totient(n): how many integers from 1 to n share a factor with n. 1 gives 0, 6 gives 4, 9 giv Read cousin_prime Whether n and n+4 are both prime. 3 pairs with 7, and 7 pairs with 11. 5 does not pair with 9. n runs from 1 t Read cows_index The first index, counting from 1, where Narayana's cow sequence equals n. 0 means n is not a term. 1 is 1, 2 i Read cows_mod Narayana's cow term a(n) modulo m. a(11) mod 5 is 3, and a(15) mod 10 is 9. n runs from 1 to 60 and m runs fro Read cows_next The smallest Narayana cow number that is at least n. 1 gives 1, 3 gives 3, 5 gives 6, and 130 gives 189. n run Read cows_sum The sum of the first n Narayana cow numbers. The first term sums to 1, the first 3 sum to 3, the first 6 sum t Read crc16 CRC-16 over text or hex bytes. Algorithms are CCITT-FALSE, ARC, XMODEM, KERMIT and MODBUS. The check string 12 Read credit_payoff Credit-card and revolving-balance payoff in exact cents. Give a balance, an annual percentage rate, and a fixe Read creditor_ref ISO 11649 creditor reference. RF, two check digits, then up to 21 letters or digits. Spaces, hyphens, dots, co Read crt Solve simultaneous congruences x ≡ a_i (mod m_i). Shared factors are allowed when the system is consistent; an Read crypto_address_validate Offline validator and chain detector for up to 50 crypto addresses: EVM (EIP-55 checksum, returns the checksum Read csp_evaluate Evaluate a Content-Security-Policy string the way CSP Evaluator does, before you ship it: parses directives, c Read css_angle Convert a CSS angle among deg, grad, turn and pirad. A degree is 1/360 turn, a gradian is 1/400 turn, and one Read css_length Convert one CSS absolute length. An inch is 96px and 2.54 cm, so a centimetre is 4800/127 px, a millimetre is Read css_resolution Convert a CSS resolution among dppx, dpi and dpcm. The alias x means dppx. One dppx is 96 dpi, and one inch is Read css_time Convert a CSS time between seconds and milliseconds. One millisecond is 1/1000 s. Minutes are not a CSS time u Read csv_profile Profile a CSV/TSV dataset: delimiter detection, row and column counts, per-column inferred type (integer, deci Read ct_subdomains Certificate Transparency subdomain enumeration via crt.sh JSON API. Returns unique hostnames (capped) issued u Read cube_sum The sum of the first n cubes, (n(n+1)/2)^2. The sum through 2 is 9, and through 10 is 3025. n runs from 0 to 1 Read cube_ways How many ways n is a sum of positive cubes, ignoring order and allowing repeats. 0 has 1 empty sum, 2 has 1, 8 Read cubefree Whether no cube divides n. 1, 12, 18, and 36 are cubefree. 8, 16, 24, and 72 are not. n runs from 1 to 1000000 Read cubefree_count How many cubefree positive integers are at most n. Through 7 there are 7, through 8 there are still 7, through Read cubefull Whether n is cubefull: 1, or every prime factor's exponent is at least 3. 1 and 8 are cubefull. 36 is powerful Read cullen The nth Cullen number, n2^n + 1. The 1st is 3, the 2nd is 9, and the 5th is 161. n runs from 1 to 256. Past 15 Read currency_format Money formatting and math with CLDR data (ICU 78 / CLDR 48, matches Intl.NumberFormat): format an amount for a Read cusip_validate CUSIP check digit. Nine characters from digits, letters, and @ . Spaces are ignored and hyphens are not. From Read cve_lookup Search NIST NVD CVE API 2.0 by product/keyword with optional CVSSv3 severity filter. Returns CVE ids, scores, Read cvss_calc Score and explain any CVSS vector: v4.0 (official FIRST macrovector algorithm, CVSS-B/BT/BE/BTE nomenclature, Read cyclotomic The cyclotomic value Phi_n(a) for an integer a at least 2. Phi_1(2) is 1, Phi_5(2) is 31, Phi_6(3) is 7, and P Read cyclotomic_coeff The coefficient of x^k in Phi_n. Phi_6 has coefficients 1, -1, 1, so k=1 is -1. The coefficient of x^7 in Phi_ Read cyclotomic_height The largest absolute coefficient of Phi_n. Phi_1 through Phi_15 have height 1. Phi_105 has height 2. n runs fr Read cyclotomic_neg Phi_n(-1). n=1 gives -2, n=2 gives 0, n=6 gives 3, and n=10 gives 5. n runs from 1 to 120. Price: $0.001 USDC Read cyclotomic_one Phi_n(1). It is 0 when n is 1. When n is a power of a prime p, it is p. Otherwise it is 1. Phi_4(1) is 2, Phi_ Read day_count Day-count conventions and accrued interest for bonds, loans and swaps: days and year fraction between two date Read dedekind_psi Dedekind's psi function, n times the product of (1+1/p) over the distinct prime factors of n. psi(1) is 1, psi Read deficiency The deficiency 2n - sigma(n). A deficient number is positive, a perfect number is 0, and an abundant number is Read delannoy_central The central Delannoy number D(n,n). D(0,0) is 1 and D(3,3) is 63. n runs from 0 to 64. Past 15 digits the valu Read depolignac Whether odd n is never a prime plus a power of two, counting 2^0. 1 qualifies, because nothing smaller works. Read depreciation Depreciation schedules: straight-line, declining balance (any factor, optional switch to straight-line), doubl Read derivative_fixed Whether n equals its arithmetic derivative. 0, 4, 27, and 3125 do. 1, 6, and 16 do not. n runs from 0 to 10000 Read digit_power_sum The sum of the digits of n each raised to k. 0 to any k from 1 up is 0. 153 to the 3 is 153. 4150 to the 5 is Read digit_rev Reverse the decimal digits of an integer. Leading zeros of the reversal are dropped, so 1200 becomes 21. The s Read digit_sum The sum of the decimal digits of an integer, ignoring the sign. 199 sums to 19. The decimal has no leading zer Read digital_root The digital root of a decimal integer. Zero stays zero, and any other magnitude is 1 + (n-1) mod 9, so multipl Read divisor_count The number of positive divisors of n. A prime has 2, and d(1) is 1. n = 0 is rejected. n runs up to 10^12. Pri Read divisor_record The largest number of divisors attained by any integer from 1 through n, and the smallest integer that attains Read divisor_sum The sum of the positive divisors of n. sigma(6) is 12 and sigma(28) is 56. n = 0 is rejected. A value past 15 Read dns_dump All common DNS records for a domain via Cloudflare DNS-over-HTTPS: A, AAAA, NS, MX, TXT, CAA. Price: $0.005 US Read dns_propagation DNS propagation check: resolves one record type for a domain across Cloudflare 1.1.1.1, Google 8.8.8.8 and Qua Read dnssec_check DNSSEC status of a domain over DNS-over-HTTPS: secure / insecure / bogus (validation failure) / island (signed Read domain_availability Bulk domain availability: for up to 20 domains, asks the authoritative registry over RDAP (server chosen from Read domain_intel One-call domain due-diligence: RDAP registration age/expiry/registrar, A/AAAA/NS/MX/CAA, live TLS certificate Read domb The Domb number D(n), the sum of C(n,k)^2 C(2k,k) C(2n-2k,n-k) for k from 0 to n. D(0) is 1, D(1) is 4, D(2) Read domb_gcd The greatest common divisor of D(n) and D(n+1). n=0 gives 1, n=1 gives 4, and n=6 gives 64. n runs from 0 to 1 Read domb_mod D(n) modulo m. D(5) mod 100 is 4, D(6) mod 1000 is 136, and D(1) mod 3 is 1. n runs from 0 to 12 and m runs fr Read domb_sum The sum of the Domb numbers D(0) through D(n). Through 0 it is 1, through 2 it is 33, through 4 it is 3005, an Read double_factorial The double factorial n!!. Even n uses the even factors down to 2, and odd n uses the odd factors down to 1. 0! Read duffinian Whether n is composite and coprime to its divisor sum. 4, 8, 9, 21, and 25 are. 1, 6, 7, and 15 are not. n run Read duffinian_count How many Duffinian numbers are at most n. Through 1 there are 0, through 4 there is 1, through 9 there are 3, Read duffinian_gcd The greatest common divisor of n and its divisor sum. 4 gives 1, 6 gives 6, 9 gives 1, and 15 gives 3. n runs Read duffinian_next The smallest Duffinian number that is at least n. 1 gives 4, 5 gives 8, 10 gives 16, and 22 gives 25. n runs f Read duffinian_sum The sum of the Duffinian numbers that are at most n. Through 4 it is 4, through 9 it is 21, through 20 it is 3 Read dyck_word Check a Dyck word of parentheses and report its semilength and height. The empty word is valid. A prefix that Read e_cf_prefix The partial quotients of e from index 0 through n, comma-separated. Through 0 it is "2". Through 5 it is "2,1, Read e_cf_sum The sum of the partial quotients of e from index 0 through n. Through 0 it is 2, through 2 it is 5, through 5 Read e_cf_term The nth partial quotient of e, starting at n=0. It is 2, then 1, 2, 1, 1, 4, 1, 1, 6. n=0 gives 2, n=5 gives 4 Read e_convergent The nth convergent of e, as numerator and denominator. n=0 is 2/1, n=1 is 3/1, n=2 is 8/3, and n=5 is 87/32. n Read e_series The partial sum of 1/k! from k=0 through n, as a reduced fraction. n=0 is 1/1, n=2 is 5/2, n=3 is 8/3, n=4 is Read ean_validate GTIN check digit for EAN-8, UPC-A (12 digits), EAN-13 and GTIN-14. The weight alternates 3 and 1 from the righ Read egcd Extended gcd of a and b. Returns g, x, and y with g = ax + by and g >= 0. gcd(0, 0) is rejected. Decimals have Read egypt_check Check that a sum of unit fractions equals the fraction a/b. A match returns the reduced sum. A sum that differ Read egypt_greedy The greedy Egyptian fraction for a proper positive fraction a/b. 5/6 is 1/2 + 1/3. A fraction that is not stri Read eip712_hash EIP-712 typed-data hashing and signature check: returns encodeType, typeHash, domain separator, struct hash an Read email_auth Email authentication & deliverability posture for a domain: SPF (qualifier, lookup count, duplicates), DMARC p Read email_validate Validate an email address without sending mail: RFC 5321/5322 syntax, MX/A lookup (null-MX aware), disposable- Read entity_extract Rule-based entity and value extraction with normalised values: dates (ISO, numeric with dmy/mdy, month names, Read envelope_fit The smallest ISO C envelope that holds a sheet flat, and whether a DL envelope holds it flat or folded in exac Read eol_check Is this software version still supported? End-of-life status from endoflife.date for 400+ products (languages, Read equation_solve Solve equations exactly where possible. Polynomials of any degree up to 100: exact rational roots, exact surd Read equidigital Whether the written factorization uses exactly as many digits as n. 1, 10, 16, and 25 do. 4 and 125 do not. n Read eu_vat_validate Validate an EU VAT number live against the European Commission VIES service: valid/invalid, plus registered tr Read euclid_mullin The nth Euclid-Mullin prime. a(1) is 2, a(4) is 43, a(7) is 5, and a(8) is 6221671. n runs from 1 to 8. Price: Read euclid_mullin_cofactor One more than the product of the earlier terms, divided by a(n). It is 1 when that Euclid number is itself pri Read euclid_mullin_index Where p sits among the first eight Euclid-Mullin primes, or 0 if it does not. 2 is 1, 43 is 4, 5 is 7, and 11 Read euclid_mullin_prod The product of the first n Euclid-Mullin primes. n=1 gives 2, n=4 gives 1806, n=7 gives 6221670, and n=8 gives Read euclid_mullin_sum The sum of the first n Euclid-Mullin primes. Through 1 it is 2, through 4 it is 55, through 7 it is 126, and t Read eui64 Modified EUI-64 from RFC 4291. A 48-bit MAC is expanded by flipping the U/L bit and inserting FF-FE in the mid Read euler_secant The nth secant number, the signed Euler number E(2n) = (-1)^n a(2n). E(0) is 1, E(2) is -1, E(4) is 5, and E( Read euler_zigzag The nth Euler zigzag number, the number of up-down permutations of n. a(0) is 1, a(3) is 2, a(4) is 5, and a(5 Read euler_zigzag_sum The sum of Euler zigzag numbers a(0) through a(n). The sum through a(5) is 26, and the sum through a(10) is 60 Read eulerian The Eulerian number A(n, k), the number of permutations of n with k ascents. A(4, 1) is 11, A(5, 2) is 66, and Read eulerian_mod A(n, k) mod m for the Eulerian numbers. The modulus is a positive decimal integer, and zero is rejected. k mus Read eulerian_poly The Eulerian polynomial A_n(x) = sum A(n, k) x^k. A_3(2) is 13, A_4(1) is 24, and A_5(0) is 1. n runs from 1 Read eulerian_row The Eulerian row A(n, 0) through A(n, n-1), comma-separated. The row for 5 is 1,26,66,26,1. It is symmetric an Read eulerian_sum The sum of Eulerian numbers A(n, k) for k from lo through hi. The whole row sums to n!. A(4, 0) through A(4, 1 Read evm_address_activity Where is this address used? One call checks an EVM address on Ethereum, Base, Arbitrum, OP Mainnet, Polygon an Read evm_allowance ERC-20 allowance matrix for an owner across up to 10 tokens × 10 spenders (defaults: major stables/WETH × Unis Read evm_balance Native ETH and ERC-20 token balances for an address on Ethereum, Base or Arbitrum via public RPC. Defaults to Read evm_block One EVM block by number, tag (latest, safe, finalized, earliest, pending) or hash on Ethereum, Base, Arbitrum, Read evm_contract_info EVM contract introspection on Ethereum, Base, Arbitrum, OP Mainnet, Polygon or BNB Chain via public RPC: bytec Read evm_gas Live gas tracker across Ethereum, Base, Arbitrum, OP Mainnet, Polygon and BNB Chain: head block age, base fee, Read evm_logs Decoded event logs for any contract and event on Ethereum, Base, Arbitrum, OP Mainnet, Polygon or BNB Chain: g Read evm_read Read-only eth_call of any view/pure function from a human signature (e.g. balanceOf(address)(uint256) or funct Read evm_token_transfers Latest ERC-20 or ERC-721 Transfer events for a token contract (optionally filtered to one wallet, in/out/both) Read evm_tx_info Decode an EVM transaction (Ethereum, Base, Arbitrum): status, block, timestamp, confirmations, from/to, value, Read evm_verify_signature Verify an Ethereum signature offline: recovers the signer address (secp256k1 ecrecover) from an EIP-191 person Read exp_count How many exponential divisors n has. 1 and 6 have 1, 8 and 12 have 2, 16 has 3, and 36 has 4. n runs from 1 to Read exp_divisors The exponential divisors of n, comma-separated and ascending. 8 gives 2,8. 12 gives 6,12. 16 gives 2,4,16. 36 Read exp_next The smallest exponential perfect number that is at least n. 1 gives 36, 37 gives 180, 180 stays 180, and 181 g Read exp_perfect Whether the exponential divisors of n sum to 2n. 36 and 180 do. 6, 12, 16, and 60 do not. n runs from 1 to 100 Read exp_sigma The sum of the exponential divisors of n. 1 gives 1, 8 gives 10, 12 gives 18, 16 gives 22, and 36 gives 72. n Read extravagant Whether the written factorization uses more digits than n. 4, 6, and 12 do. 10 and 125 do not. n runs from 1 t Read factor_concat One concatenation of the prime factors of n, repeats included and primes left unchanged. 2 gives 2, 4 gives 22 Read factor_digits How many digits the written prime factorization uses, omitting exponents of 1. 1 uses 1, 4 uses 2, 12 uses 3, Read factoradic The factorial-base digits of n, from the highest place down through the 0! place, comma-separated. The last di Read factoradic_decode The integer value of factorial-base digits. The last entry is the 0! place. [2, 1, 0] is 5 and [3, 2, 1, 0] is Read factoradic_digit The digit of the k! place of n. It is floor(n / k!) mod (k+1), and it is 0 when k! exceeds n. The 0! digit is Read factoradic_length How many factorial-base digits n has, including the 0! place. 0 has 1, 5 has 3, 23 has 4, and 24 has 5. n runs Read factoradic_sum The sum of the factorial-base digits of n. 0 sums to 0, 5 sums to 3, 23 sums to 6, and 463 sums to 9. n runs f Read factorial_mod n! mod m. 0! is 1, so 0! mod 5 is 1. 10! mod 97 is 30. The modulus is a positive decimal integer, and zero is Read factorial_prime Whether n! + 1 and n! - 1 are prime. Both are prime for n=3. For n=11, only n! + 1 is prime. For n=12, only n! Read factorial_valuation The p-adic valuation of n!, the exponent of the prime p in n!. v_2(10!) is 8, v_3(10!) is 4, and v_2(100!) is Read factorion Whether n equals the sum of the factorials of its digits. 1, 2, 145, and 40585 do. 24 and 10 do not. n runs fr Read fake_data Generate realistic, reproducible fake test data from a field schema: up to 1,000 records with 44 field types ( Read falling The falling factorial (n)_k = n(n-1)...(n-k+1). (5)_3 is 60, (5)_0 is 1, and (5)_6 is 0. n runs from 0 to 1000 Read farey_adjacent Whether a/b and c/d are Farey neighbors. They are adjacent when the determinant ad-bc is 1 or -1. A non-positi Read farey_child The mediant child of a Farey-neighbor pair. a/b and c/d must satisfy |ad-bc| = 1. A pair that is not adjacent Read farey_count How many reduced fractions lie in the Farey sequence of order n, from 0/1 through 1/1. Order 1 has 2, order 5 Read fda_adverse_events FDA adverse-event reports (openFDA): drug FAERS, device MAUDE, food/supplement CAERS. Filter by drug, product, Read fda_device FDA device records (openFDA): 510(k) clearances, PMA approvals, device classification (product code, class, re Read fda_recalls FDA recalls (openFDA): drug, device and food enforcement reports (class, status, firm, reason, distribution, l Read feed_discover Find a website's RSS/Atom/JSON feeds and read the latest items: parses <link rel="alternate"> tags, likely fee Read fermat_number The Fermat number 2^(2^n) + 1. n=0 gives 3, n=4 gives 65537, and n=5 gives 4294967297. n runs from 0 to 12. Pa Read fermat_prime Whether the Fermat number 2^(2^n) + 1 is prime. n from 0 through 4 gives a prime. n=5 does not: 4294967297 = 6 Read fibbinary Whether the binary expansion of n has no two adjacent 1 bits. 0, 1, 2, 5, 10, and 21 do. 3, 6, and 7 do not. n Read fibbinary_count How many fibbinary numbers from 0 through n there are. Through 0 there is 1, through 2 there are 3, through 3 Read fibbinary_index The position of n among the fibbinary numbers, counting from 1, or 0 if n is not fibbinary. 0 is 1, 1 is 2, 5 Read fibbinary_next The smallest fibbinary number that is at least n. 0 stays 0, 3 gives 4, 6 and 7 give 8, 11 gives 16, and 21 st Read fibbinary_nth The nth fibbinary number, counting from 1, with 0 in position 1. The 1st is 0, the 4th is 4, the 8th is 10, th Read fibword The nth bit of the infinite Fibonacci word, starting at n = 0. The word is the fixed point of 0 to 01 and 1 to Read fibword_prefix The first n bits of the Fibonacci word. The first 8 bits are 01001010, and the first 16 bits are 0100101001001 Read fifth_sum The sum of the first n fifth powers, n^2(n+1)^2(2n^2+2n-1)/12. The sum through 10 is 220825. n runs from 0 to Read figi_validate FIGI check digit. Twelve characters with no vowels. The first two are letters and are not BS, BM, GG, GB or VG Read fine The nth Fine number. F(0) is 1, F(1) is 0, F(2) is 1, and F(6) is 57. For n at least 1, F(n) = C(n) - sum of C Read fine_sum The sum of Fine numbers from lo through hi inclusive. The sum from F(0) through F(6) is 85. lo must not pass h Read fitness_calc Body and training calculator in one call, metric or imperial: BMI with WHO category and healthy-weight range, Read fletcher16 Fletcher-16 checksum, modulus 255. The two sums are packed with the second sum in the high byte. "abcde" is c8 Read fletcher32 Fletcher-32 checksum, modulus 65535. Bytes are big-endian 16-bit words. An odd final byte is padded with zero. Read float_bits Explain an IEEE 754 binary32 or binary64 value. Pass bits as hex, or a decimal or fraction, which is rounded t Read float_next The next binary32 or binary64 value up or down. Positive zero and negative zero both step up to the smallest p Read float_order IEEE 754 totalOrder on two binary32 or binary64 values. Negative zero is below positive zero. Negative NaNs ar Read float_ulp Unit in the last place of a binary32 or binary64 value. A power of two uses the spacing of its own binade. Zer Read food_barcode Food product facts by barcode (EAN-13, EAN-8, UPC-A, UPC-E; GTIN check digit validated) from Open Food Facts ( Read four_squares Four nonnegative integers w <= x <= y <= z with w^2 + x^2 + y^2 + z^2 = n. Every nonnegative integer has one. Read fourth_free Whether no fourth power divides n. 1, 8, 12, and 36 qualify. 16, 32, 48, and 81 do not. n runs from 1 to 10000 Read fourth_sum The sum of the first n fourth powers, n(n+1)(2n+1)(3n^2+3n-1)/30. The sum through 0 is 0, through 1 is 1, and Read franel The Franel number F(n), the sum of C(n,k)^3 for k from 0 to n. F(0) is 1, F(1) is 2, F(2) is 10, F(4) is 346, Read franel_central The middle summand of the Franel number, C(n, floor(n/2))^3. n=0 gives 1, n=2 gives 8, n=4 gives 216, n=5 give Read franel_gcd The greatest common divisor of F(n) and F(n+1). n=0 gives 1, n=1 gives 2, and n=6 gives 16. n runs from 0 to 1 Read franel_mod F(n) modulo m. F(4) mod 7 is 3, F(5) mod 10 is 2, and F(8) mod 100 is 62. n runs from 0 to 16 and m runs from Read franel_sum The sum of the Franel numbers F(0) through F(n). Through 0 it is 1, through 1 it is 3, through 4 it is 415, an Read frobenius The Frobenius number of two coprime integers a and b, each at least 2. It is ab-a-b, the largest integer that Read frugal Whether the written factorization uses fewer digits than n. 125 and 128 do. 1, 16, and 12 do not. n runs from Read frugal_count How many frugal numbers are at most n. Through 124 there are none, through 125 there is 1, and through 200 the Read fubini_mod The nth Fubini number mod m. The modulus is a positive decimal integer, and zero is rejected. n runs from 0 to Read fubini_sum The sum of Fubini numbers from lo through hi inclusive. The sum from a(0) through a(5) is 634. lo must not pas Read fund_holdings Mutual fund / ETF portfolio from its latest public SEC Form N-PORT: top holdings by value (name, CUSIP, ISIN, Read fuss_catalan The Fuss-Catalan number C(pn+1, n) / (pn+1). FC(2,n) is the Catalan number, so FC(2,5) is 42. p runs from 1 to Read fuss_check The Fuss-Catalan division. num is C(pn+1, n), den is pn+1, and value is the exact quotient. p runs from 1 to 8 Read fuzzy_match Fuzzy string matching for entity resolution, dedupe and "did you mean": match one or many queries against up t Read fx_rate Fiat currency exchange rate (ECB reference rates via Frankfurter, with ExchangeRate-API fallback for 150+ curr Read gcd_sum The sum of gcd(x, y) over 1 <= x,y <= n. Through 1 it is 1, through 5 it is 37, and through 10 it is 189. n ru Read genocchi The nth Genocchi number |G(2n)|, from G(2n) = 2(1-4^n)B(2n) with B(1) = +1/2. The first four absolute values a Read geo_polygon Analyse GeoJSON polygons (Polygon, MultiPolygon, Feature or FeatureCollection): geodesic area (m², km², ha, ac Read geocode Geocode an address or place name to latitude/longitude (US Census Geocoder for US street addresses, Photon/Ope Read github_health GitHub repository health summary via public API: stars/forks/issues, license, languages, last push age, latest Read github_trending Trending new GitHub repositories: repos created in the last day, week or month ranked by stars (GitHub REST se Read golay_rudin The Golay-Rudin-Shapiro partial sum. Each term is +1 or -1 as the Rudin-Shapiro sign of k, and the value is th Read goldbach The Goldbach split of an even n with the smallest prime p, so n = p+q and p <= q. 10 gives 3 and 7. 20 gives 3 Read goldbach_check Whether p and n-p are both prime. For n=10, p=3 and p=5 work, and p=2 does not. n is even, from 4 to 10000, an Read goldbach_count How many Goldbach splits an even n has, with p <= q. 10 has 2, 20 has 2, and 100 has 6. n is even, from 4 to 1 Read golomb The nth Golomb term, 1-based. a(1) is 1, a(n) counts how often n will appear, and a(6) is 4. The sequence star Read golomb_inverse The smallest index n with Golomb a(n) equal to k. Index 1 holds 1, index 4 holds 3, and index 6 holds 4. k run Read golomb_sum The sum of the first n Golomb terms, from a(1) through a(n). The sum of the first 5 terms is 11. n runs from 1 Read gray_bits The binary reflected Gray code of n, written in exactly width bits. Bit 0 is on the right. n must be smaller t Read gray_decode Decode a binary reflected Gray code back to its index. The walk xors the code with each of its right shifts. A Read gray_encode Encode a non-negative integer as a binary reflected Gray code. The code is n xor (n shifted right by one). Zer Read gray_flip The single bit that flips in the binary reflected Gray code when the index goes from n to n+1. Bit 0 is the le Read gray_next The next binary reflected Gray code after g. It is the code of the index one past the index of g. A negative c Read hamming Whether n is a Hamming number, that is 5-smooth. 1, 10, 30, and 36 are. 7 and 14 are not. n runs from 1 to 100 Read hamming_count How many Hamming numbers are at most n. Through 1 there is 1, through 7 there are 6, through 10 there are 9, t Read hamming_index Where n sits in the Hamming sequence, counting from 1, or 0 if it is not Hamming. 1 is 1, 12 is 10, 36 is 20, Read hamming_next The smallest Hamming number that is at least n. 7 gives 8, 10 stays 10, 11 gives 12, and 31 gives 32. n runs f Read hamming_nth The nth Hamming number, counting from 1. The 1st is 1, the 10th is 12, the 18th is 30, and the 20th is 36. n r Read happy Follow the happy-number map: replace n by the sum of the squares of its decimal digits. 1 is already happy. 7 Read harmonic The nth harmonic number as a reduced fraction. H1 is 1/1, H2 is 3/2, and H6 is 49/20. n < 1 is rejected. n run Read harshad Whether n is a base-10 Harshad number: n is divisible by the sum of its decimal digits. 18 is Harshad, and 11 Read harshad_base Whether n is a Harshad number in base b: n is divisible by the sum of its base-b digits. 10 is Harshad in base Read hash_encode Hashing, HMAC and encoding toolkit: digests (MD5, SHA-1/2/3, Keccak-256 as used by Ethereum, RIPEMD-160, Whirl Read headers_audit Graded HTTP security-header audit of a URL: HSTS, CSP quality, XFO, nosniff, Referrer/Permissions/COOP/CORP/CO Read hermite The physicist Hermite polynomial H(n, x) at an integer x. H(0, x) is 1, H(1, x) is 2x, H(2, 1) is 2, and H(3, Read hermite_deriv The derivative of the physicist Hermite polynomial, H'(n, x) = 2nH(n-1, x). At n = 3 and x = 1 the derivative Read hermite_prob The probabilist Hermite polynomial He(n, x) at an integer x. He(0, x) is 1, He(1, x) is x, and He(2, 1) is 0. Read heron_area Exact area data for a strict integer triangle. sixteen is 16 times the squared area. When sixteen is a perfect Read hexanacci The nth hexanacci number. H(0) through H(4) are 0, H(5) is 1, and H(10) is 16. Each later term adds the previo Read hexanacci_index The smallest index n with hexanacci number H(n) equal to value. 0 is index 0, and 1 is index 5. A number that Read hexanacci_mod H(n) mod m for the hexanacci numbers. H(10) mod 7 is 2, and H(57) mod 97 is 61. The modulus is a positive deci Read hexanacci_step The hexanacci step H(n) = H(n-1) + H(n-2) + H(n-3) + H(n-4) + H(n-5) + H(n-6). a through f are those six earli Read hexanacci_sum The sum of hexanacci numbers from lo through hi. The sum from H(0) through H(10) is 32. lo must not pass hi. E Read higgs Whether n is a Higgs prime for exponent 2. 2, 13, and 19 are. 4, 9, and 17 are not. n runs from 1 to 2000. Pri Read higgs_count How many Higgs primes for exponent 2 are at most n. Through 1 there are 0, through 16 there are 6, through 19 Read higgs_gap The distance from a Higgs prime to the next one. 2 gives 1, 7 gives 4, 13 gives 6, and 71 gives 8. A number th Read higgs_index The 1-based place of n in the Higgs primes for exponent 2, or 0 when n is absent. 2 is place 1, 19 is place 7, Read higgs_next The smallest Higgs prime for exponent 2 that is at least n. 1 gives 2, 14 gives 19, 17 gives 19, and 73 gives Read highly_abundant Whether sigma(n) is strictly larger than sigma(m) for every m below n. 1, 6, 8, and 16 are. 5 and 9 are not. n Read highly_composite Whether n is highly composite: it has strictly more divisors than every smaller positive integer. 1, 2, 6, 12, Read highly_composite_count How many highly composite numbers are from 1 through n. There are 4 through 10, 9 through 60, and 10 through 1 Read highly_composite_next The least highly composite number greater than or equal to n. 1 gives 1, 3 gives 4, 7 gives 12, and 5041 gives Read hn_search Search Hacker News stories and comments (Algolia HN index): by relevance or newest first, with date range, sto Read hoax Whether n is a composite hoax number: its digit sum equals the digit sum of its distinct prime factors. 22, 58 Read hof_conway The nth Hofstadter-Conway term. a(1) and a(2) are 1, and a(n) = a(a(n-1)) + a(n-a(n-1)). a(5) is 3, and a(10) Read hof_conway_sum The sum of Hofstadter-Conway terms a(1) through a(n). The sum through a(10) is 32, and the sum through a(30) i Read hof_female The nth Hofstadter female term. F(0) is 1, M(0) is 0, F(n) = n - M(F(n-1)), and M(n) = n - F(M(n-1)). F(1) is Read hof_g The nth Hofstadter G term. G(0) is 0, and G(n) = n - G(G(n-1)). G(1) is 1, and G(10) is 6. n runs from 0 to 10 Read hof_h The nth Hofstadter H term. H(0) is 0, and H(n) = n - H(H(H(n-1))). H(1) is 1, and H(10) is 7. n runs from 0 to Read hof_male The nth Hofstadter male term. M(0) is 0, F(0) is 1, M(n) = n - F(M(n-1)), and F(n) = n - M(F(n-1)). M(2) is 1, Read hof_pair The Hofstadter male and female terms at the same index. M(0) is 0 and F(0) is 1. At n = 5 both terms are 3. n Read hof_q The nth Hofstadter Q term. Q(1) and Q(2) are 1, and Q(n) = Q(n-Q(n-1)) + Q(n-Q(n-2)). Q(10) is 6, and Q(16) is Read hof_q_prefix The first n Hofstadter Q terms, from Q(1) through Q(n), written as comma-separated integers. The first 10 term Read hof_q_sum The sum of Hofstadter Q terms Q(1) through Q(n). The sum through Q(10) is 36, and the sum through Q(20) is 129 Read hofstadter Hofstadter G. G(0) is 0, G(1) and G(2) are 1, G(5) is 3, and G(10) is 6. The rule is G(n) = n - G(G(n-1)). n r Read hofstadter_conway Conway's Hofstadter sequence. a(1) and a(2) are 1, a(6) is 4, a(10) is 6, and a(16) is 8. The rule is a(n) = a Read hofstadter_f Hofstadter's female sequence. F(0) and F(1) are 1, F(5) is 3, and F(7) and F(8) are 5. The male term at the sa Read hofstadter_m Hofstadter's male sequence. M(0) and M(1) are 0, M(2) is 1, M(5) is 3, and M(7) is 4. n runs from 0 to 5000. P Read hofstadter_sum The sum of Hofstadter G from 0 through n. Through 0 it is 0, through 1 it is 1, through 5 it is 10, and throug Read home_grows Whether the home prime of n has more digits than n. 4, 6, and 10 do. 2 and 21 do not. A chain past 20 steps or Read home_prime The home prime of n. 2 stays 2, 4 becomes 211, 10 becomes 773, and 16 becomes 31636373. The search stops after Read home_prime_len How many digits the home prime of n has. 2 has 1, 4 and 10 have 3, and 16 has 8. A chain past 20 steps or 12 d Read home_prime_steps How many factor-concatenations turn n into a prime. 2 takes 0, 6 and 12 take 1, 4 takes 2, and 10 and 16 take Read hsts_preload HSTS preload check for a domain: current status on the Chromium preload list (hstspreload.org), the official p Read html_sanitize Sanitize untrusted HTML (LLM output, user comments, scraped pages, email bodies) with a strict allowlist befor Read html_tables Extract HTML tables from a public URL (SSRF-safe fetch) or an HTML string into clean JSON: header detection (t Read html_to_markdown Convert an HTML string to clean Markdown (GFM tables, fenced code, nested lists). No fetch. Body: { "html": ". Read http_probe Fast HTTP GET probe: final URL, status, timing, response bytes, and a subset of interesting headers (server, c Read hyperperfect Whether n = 1 + k(sigma(n) - n - 1) for a positive integer k, and that k. 6 and 28 give k=1. 21 gives k=2. 12 Read hypothesis_test Run a statistical significance test and get the statistic, p-value, effect size, confidence interval and a pla Read iban_validate Validate an IBAN offline: ISO 7064 mod-97 checksum, per-country length and BBAN structure from the SWIFT IBAN Read icbrt The integer cube root of a non-negative decimal integer, the greatest r with r^3 not exceeding n. icbrt(26) is Read identicon_svg Deterministic avatar SVG for any user id, email, wallet address or name — the same input always gives the same Read image_info Inspect an image without decoding pixels: format, dimensions (EXIF-orientation aware), megapixels, animation, Read imei_validate IMEI and IMEISV. A 15-digit IMEI must pass the Luhn check. A 14-digit IMEI has no check digit, and a 16-digit Read impermanent_loss Impermanent loss calculator for AMM liquidity positions: full-range constant-product pools (Uniswap v2, Raydiu Read inet_checksum RFC 1071 internet checksum. Big-endian 16-bit words are added with end-around carry, then inverted. An odd fin Read infinitary_abundant Whether the infinitary divisors of n sum to more than 2n. 24 and 30 do. 6, 16, 36, and 48 do not. n runs from Read infinitary_count How many infinitary divisors n has. 6 and 8 and 12 have 4, 16 has 2, and 24 has 8. n runs from 1 to 1000000. P Read infinitary_deficient Whether the infinitary divisors of n sum to less than 2n. 1, 16, 36, and 48 do. 6, 24, and 60 do not. n runs f Read infinitary_perfect Whether the infinitary divisors of n sum to 2n. 6, 60, and 90 do. 16, 24, 36, and 48 do not. n runs from 1 to Read infinitary_sigma The sum of the infinitary divisors of n. 6 gives 12, 8 gives 15, 16 gives 17, 36 gives 50, and 48 gives 68. n Read invoice_total Invoice math in exact cents. Each line is quantity × unit price, then an optional line discount, rounded half Read involution The number of involutions of n objects. I(0) and I(1) are 1, and I(n) = I(n-1) + (n-1)I(n-2). I(5) is 26, and Read involution_fixed The number of involutions of n objects with exactly k fixed points. It is 0 when n-k is odd. For n = 5 and k = Read involution_mod I(n) mod m, the involution number reduced by a positive modulus. Zero is rejected. n runs from 0 to 100. The r Read involution_sum The sum of involution numbers from lo through hi inclusive. The sum from I(0) through I(10) is 13232. lo must Read ioc_extract Extract indicators of compromise from threat reports, alerts, logs or emails: IPv4/IPv6 (private ranges option Read ip_intel IP ASN + geolocation from free public sources (ipapi.co + RIPE Stat prefix-overview). Country/city/coords, ASN Read ipow The integer power base^exp. 2^10 is 1024, and (-2)^5 is -32. 0^0 is 1. exp runs from 0 to 256. The base has at Read isbn_validate ISBN-10 and ISBN-13 check digit. Spaces and hyphens are ignored, and a 9-digit number is read as an ISBN-10 wi Read isin_validate ISIN check digit (ISO 6166). Two letters, nine letters or digits, one check digit. Letters expand to their alp Read isqrt The integer square root of a non-negative decimal integer, the greatest r with r^2 not exceeding n. isqrt(0) i Read isquare Whether a non-negative decimal integer is a perfect square, and the root when it is. 16 is 4 squared. 15 is no Read issn_validate ISSN check digit (8 characters, the last may be X). Spaces and hyphens are ignored. The formatted form is the Read jacobi The Jacobi symbol (a/n). n must be odd and positive. The value is -1, 0, or 1. An even or non-positive n is re Read jacobsthal The nth Jacobsthal number. J0 is 0, J1 is 1, and each later term is the previous one plus twice the one before Read jacobsthal_lucas The nth Jacobsthal-Lucas number. L0 is 2, L1 is 1, and later terms use the Jacobsthal recurrence. An optional Read jacobsthal_power The power identity J(n) + J(n+1) = 2^n. left is J(n), right is J(n+1), and value is their sum. n runs from 0 t Read jacobsthal_sum The sum of Jacobsthal numbers, or Jacobsthal-Lucas numbers, from lo through hi inclusive. seq is j or l. lo mu Read jordan_totient The Jordan totient J(k, n). J(1, n) is Euler's totient. J(1, 6) is 2 and J(2, 6) is 24. k runs from 1 to 12 an Read josephus_first The first person eliminated in the Josephus circle. With 5 people and every 2nd, person 2 leaves first. With e Read josephus_order The elimination order in the Josephus circle, comma-separated, not including the survivor. With 5 people and e Read josephus_place When a seat is eliminated. Place 1 is the first person out. The survivor has place n. With 5 people and every Read json_canonicalize RFC 8785 JSON Canonicalization Scheme (JCS): deterministic serialization with keys sorted by UTF-16 code units Read json_diff Structural diff of two JSON documents: list of added / removed / changed values with JSON Pointer paths and ol Read json_repair Repair broken or LLM-style JSON into valid JSON: strips markdown fences and surrounding prose, comments, trail Read json_schema_infer Infer a JSON Schema (draft 2020-12) from one JSON document or many samples: types (integer vs number), nested Read json_to_types Generate typed models from sample JSON: TypeScript interfaces, Zod schemas, Python Pydantic models, Go structs Read jsonpath Query JSON with JSONPath (RFC 9535): dot and bracket names, indexes, slices, wildcards, recursive descent (..) Read juggler_chain The juggler path from n through 1, comma-separated. 1 gives "1" and 3 gives "3,5,11,36,6,2,1". n runs from 1 t Read juggler_max The largest term on the juggler path from n down to 1. 1 peaks at 1, 3 peaks at 36, and 9 peaks at 140. n runs Read juggler_odd How many odd terms appear before the final 1 on the juggler path. 1 and 2 have none, 7 has 1, 5 has 2, and 3 a Read juggler_step One juggler step. An even n goes to its integer square root, and an odd n goes to the integer square root of i Read juggler_steps How many juggler steps reach 1. 1 takes 0, 2 takes 1, 3 takes 6, 7 takes 4, and 9 takes 7. n runs from 1 to 20 Read jwt_decode Decode a JWT (JWS or JWE header), POST only (token in the JSON body, never in a URL): header, claims, alg/kid, Read k8s_manifest_lint Security and reliability audit of Kubernetes YAML (multi-document, List kinds; kubesec / kube-score style): pr Read kaprekar_chain The four-digit Kaprekar chain from n through 6174, comma-separated. 3524 gives "3524,3087,8352,6174" and 6174 Read kaprekar_kernel The value the four-digit Kaprekar routine reaches. Every non-repdigit from 0 to 9999 reaches 6174. A repdigit Read kaprekar_repdigit Whether the four-digit padding of n uses only one digit. 0, 1111, and 2222 do. 5, 111, and 6174 do not. n runs Read kaprekar_step One step of the four-digit Kaprekar routine, padding with leading zeros. 3524 gives 3087, 14 gives 4086, 6174 Read kaprekar_steps How many steps the four-digit Kaprekar routine takes to reach 6174. 6174 takes 0, 3524 takes 3, 5 takes 6, and Read keith Whether n is a Keith number. 14, 19, 28, 47, 61, 75, and 197 are. 10, 11, and any one-digit number are not. n Read keith_count How many Keith numbers are at most n. Through 13 there are none, through 14 there is 1, through 100 there are Read keith_next The smallest Keith number that is at least n. 1 and 14 give 14, 15 gives 19, and 76 gives 197. n runs from 1 t Read keith_row The digit recurrence of a Keith number, comma-separated, through n itself. 14 gives "1,4,5,9,14". A number tha Read keith_steps How many new terms are generated after the digits before a Keith number reaches itself. 14 takes 3 and 197 tak Read keyword_extract Keyphrase extraction for tagging, SEO and RAG metadata: RAKE (rapid automatic keyword extraction: candidate ph Read kolakoski The nth Kolakoski term, using a 1-based index. The sequence is 1s and 2s, and it starts 1, 2, 2, 1, 1. Term 7 Read kolakoski_check Whether a string of the digits 1 and 2 is a prefix of the Kolakoski sequence. 122112 is a prefix. An empty str Read kolakoski_prefix The first n Kolakoski terms as a string of the digits 1 and 2. The first 10 terms are 1221121221. n runs from Read kolakoski_sum The sum of the first n Kolakoski terms. The sum of the first 10 terms is 15. n runs from 1 to 100000. Price: $ Read kurepa The Kurepa function K(n), the sum of k! for k from 0 through n-1. K(1) is 1, K(2) is 2, and K(5) is 34. This i Read kurepa_mod K(n) mod m for the Kurepa function, the sum of k! from 0 through n-1. The modulus is a positive decimal intege Read kynea The nth Kynea number, (2^n + 1)^2 - 2. n=1 gives 7, n=2 gives 23, and n=3 gives 79. n runs from 1 to 128. Past Read laguerre The Laguerre polynomial L(n, x) at an integer x, as a reduced fraction. L(n, 0) is 1, L(1, 1) is 0, and L(2, 1 Read lah_row Row n of the unsigned Lah numbers, from L(n, 0) through L(n, n), written as comma-separated integers. Row 5 is Read landau Landau's function g(n), the largest least common multiple of the parts of a partition of n. g(0) is 1, g(5) is Read landau_index The smallest n with Landau value g(n) equal to value. 1 is index 0, and 6 is index 5. A number that is not a v Read landau_mod g(n) mod m for Landau's function. g(12) mod 7 is 4, and g(25) mod 97 is 96. The modulus is a positive decimal Read landau_sum The sum of Landau's function from lo through hi. The sum from g(0) through g(10) is 100. lo must not pass hi. Read largest_prime_factor The largest prime factor of n. It is 1 for 1, 3 for 12, 5 for 30, and 7 for 35. n runs from 1 to 1000000000000 Read lazy_caterer The lazy caterer number, the most pieces of a pancake cut by n straight cuts: n(n+1)/2 + 1. Zero cuts give 1, Read least_primitive_root The least primitive root modulo the prime p. It is 1 for 2, 3 for 7, 5 for 23, and 6 for 41. p must be prime a Read legendre_pol The Legendre polynomial P(n, x) at an integer x, as a reduced fraction. P(n, 1) is 1, and P(2, 2) is 11/2. n r Read legendre_symbol The Legendre symbol (a/p) for an odd prime p. (2/7) is 1, (3/7) is -1, and (-1/5) is 1. p must be an odd prime Read lei_validate LEI check digits, ISO 17442 with ISO 7064 mod 97-10. Twenty letters or digits, and the whole number is 1 mod 9 Read leonardo The nth Leonardo number. L(0) and L(1) are 1, and L(n) = L(n-1) + L(n-2) + 1. L(4) is 9, and L(10) is 177. An Read leonardo_fib The Leonardo identity L(n) = 2F(n+1) - 1. F is the Fibonacci sequence with F(0) = 0 and F(1) = 1. For n = 10, Read leonardo_index The smallest index n with L(n) equal to value. 1 is index 0, and 9 is index 4. A number that is not a Leonardo Read leonardo_mod L(n) mod m for the Leonardo sequence. The modulus is a positive decimal integer, and zero is rejected. n runs Read leonardo_sum The sum of Leonardo numbers from lo through hi inclusive. The sum from L(0) through L(5) is 34. lo must not pa Read levy Levy writes an odd n as p+2q with p and q prime, choosing the smallest p. 7 is 3+22, 27 is 5+211, and 21 is 7+ Read license_detect Identify a software / content licence from a LICENSE file, source header or SPDX expression and explain what i Read link_check Find broken links on a public web page (or in supplied HTML, e.g. a draft before publishing): fetches the page Read liouville Liouville's function lambda(n) = (-1) to the number of prime factors of n, counted with multiplicity. lambda(1 Read liouville_sum The sum of Liouville's lambda(k) from 1 through n. The sum through 10 is 0, and the sum through 30 is -4. n ru Read liquidation_price Liquidation calculator. mode=perp: isolated-margin long/short liquidation and bankruptcy price from entry pric Read little_omega The number of distinct prime factors of n. omega(1) is 0, omega(4) is 1, and omega(30) is 3. n runs from 1 to Read llms_txt Read a site's llms.txt (the llmstxt.org convention that tells AI agents where the useful docs are), SSRF-safe: Read loan_amortization Loan / mortgage amortization: periodic payment, total interest, payoff date and a yearly or full payment sched Read loan_apr True cost of a loan including fees: the US Truth in Lending APR (Regulation Z actuarial method) and the EU APR Read look_say_count How many times a digit occurs in the nth look-and-say term. Term 5 is 111221, so it has four 1s and two 2s. A Read look_say_digit One digit of the nth look-and-say term. The index starts at 0. Term 5 is 111221, so index 3 is 2. An index pas Read look_say_len The number of digits in the nth look-and-say term, starting from 1. Term 1 has length 1, term 5 has length 6, Read look_say_step One look-and-say step on a digit string. 1 becomes 11, and 1211 becomes 111221. Two 2s become 22. An empty str Read lucas The nth Lucas number. L0 is 2, L1 is 1, and each later term is the sum of the previous two. L5 is 11 and L10 i Read lucas_disc The Lucas discriminant L(n)^2 - 5F(n)^2, which equals 4(-1)^n. F is the Fibonacci sequence with F0 = 0 and F1 Read lucas_index The smallest index n with L(n) equal to value. 2 is index 0, and 1 is index 1. A number that is not a Lucas nu Read lucas_mod L(n) mod m. The modulus is a positive decimal integer, and zero is rejected. n runs from 0 to 4096. The residu Read lucas_sum The sum of Lucas numbers from lo through hi inclusive. The sum from L0 through L5 is 28. The sum equals L(hi+2 Read lucky Whether n is a lucky number. The list begins 1, 3, 7, 9, 13, 15, 21. 5 and every even number are not lucky. n Read lucky_count How many lucky numbers are from 1 through n. There is 1 through 1, 2 through 3, and 13 through 50. n runs from Read lucky_next The least lucky number greater than or equal to n. 2 gives 3, 20 gives 21, and 49 gives 49. n runs from 1 to 9 Read lucky_nth The kth lucky number. The 1st is 1, the 10th is 33, and the 20th is 79. k runs from 1 to 8772, which is the co Read lynch_bell Whether every digit of n is nonzero, the digits are all different, and each digit divides n. 12, 15, 24, 124, Read mac_address Normalize a MAC-48 or EUI-64. Accepts colon, hyphen, Cisco dotted (groups of four) or plain hex, any case. Ret Read macro_indicators Macro indicators by country and year: GDP, real GDP growth, GDP per capita, inflation, unemployment, populatio Read markdown_to_html Render Markdown to safe HTML: CommonMark-style blocks (ATX / setext headings, paragraphs, block quotes, nested Read math_eval Evaluate math expressions safely and exactly — the calculator agents should call instead of doing arithmetic i Read matrix_ops Linear algebra on matrices, exactly where it matters. Determinant, inverse, RREF, rank, null space, linear sol Read mcnugget Whether n nuggets can be ordered in boxes of 6, 9, and 20. 0, 6, and 44 can. 1, 11, and 43 cannot. n runs from Read mcnugget_count How many integers from 0 through n can be ordered in boxes of 6, 9, and 20. Through 0 there is 1, through 6 th Read mcnugget_miss How many positive integers through n cannot be ordered in boxes of 6, 9, and 20. Through 1 there is 1, through Read mediant The mediant of a/b and c/d, which is (a+c)/(b+d) reduced by their gcd. Numerators may be zero. A non-positive Read meid_validate MEID, the CDMA equipment id. Hex is 14 digits with an optional Luhn check in base 16. Decimal is 18 digits (ma Read mersenne The Mersenne number 2^n - 1. Exponent 1 gives 1, 2 gives 3, and 31 gives 2147483647. n runs from 1 to 128. Pas Read mersenne_prime Whether 2^n - 1 is prime. Exponents 2, 3, 5, 7, 13, 17, 19, and 31 are, inside this cap. Exponent 11 is not, b Read mertens The Mertens function M(n), the sum of the Mobius function mu(k) from 1 through n. M(1) is 1, M(10) is -1, and Read mertens_max The largest absolute value of the Mertens function M(k) for k from 1 through n, and the smallest k where it oc Read mex The mex of a list: the smallest missing nonnegative integer. The mex of 0, 1, 3 is 2. The mex of 1, 2, 3 is 0. Read midi_note A written note or a MIDI number. Middle C is C4, which is 60. C-1 is 0 and G9 is 127. The name you pass keeps Read mobius The Mobius function mu(n). It is 0 when n has a squared prime factor, and otherwise (-1)^k for k distinct prim Read mod_inv The modular inverse of a modulo m, as a residue from 0 up to m-1. The modulus must be at least 1. A value that Read mod_inverse The inverse of a modulo m, the residue from 0 to m-1 with ainverse = 1 mod m. The inverse of 3 mod 7 is 5, and Read mod_pow base^exp mod m, exact, for a non-negative exponent. The modulus must be at least 1. A negative exponent is rej Read moon_phase Moon phase for any moment: phase name and emoji, illuminated fraction, phase angle, age in days since new moon Read moran Whether n divided by its digit sum is a prime. 18, 21, and 27 are. 1, 10, 12, and 24 are not. n runs from 1 to Read moran_count How many Moran numbers are at most n. Through 1 there are none, through 18 there is 1, through 21 there are 2, Read moran_next The smallest Moran number that is at least n. 1 and 18 give 18, 19 gives 21, 22 gives 27, and 100 gives 111. n Read moran_ratio n divided by its digit sum, when that division is exact. 18 gives 2, 21 gives 7, 12 gives 4, and 1 gives 1. A Read moran_sum The sum of the Moran numbers that are at most n. Through 1 it is 0, through 18 it is 18, through 21 it is 39, Read morse_decode Decode international Morse. Letters are separated by one space and words by " / ". The result is capitals. ... Read morse_encode Encode text as international Morse (ITU-R M.1677). Letters are separated by one space and words by " / ". SOS Read morse_timing How long a Morse message takes at a given speed. The dit is 1200/wpm milliseconds. PARIS is 43 units, or 50 un Read mortgage_affordability How much house can I afford: maximum home price and loan from income, monthly debts, down payment and rate und Read moser The nth Moser-de Bruijn number, binary digits of n read in base 4. n=0 gives 0, n=3 gives 5, n=5 gives 17, n=6 Read moser_check Whether every base-4 digit of n is 0 or 1. 0, 1, 4, 5, and 20 qualify. 2 and 6 do not. n runs from 0 to 100000 Read moser_count How many Moser-de Bruijn numbers are at most n, counting 0. Through 0 there is 1, through 1 there are 2, throu Read moser_index The index whose Moser-de Bruijn number is n. 0 is index 0, 5 is index 3, 20 is index 6, and 64 is index 8. A b Read moser_sum The sum of the Moser-de Bruijn numbers a(0) through a(n). Through 0 it is 0, through 3 it is 10, through 5 it Read motzkin The nth Motzkin number. M0 and M1 are 1, M2 is 2, M4 is 9, from M_n = sum binom(n, 2k) C_k. n runs from 0 to 5 Read mult_order The multiplicative order of a modulo n: the least positive k with a^k = 1 mod n. The order of 2 mod 7 is 3, an Read multiperfect Whether sigma(n) is a multiple of n, with multiplicity k at least 2. 6 and 28 give k=2. 120 and 672 give k=3. Read multiplicative_persistence Multiplicative persistence: how many times the decimal digits of n must be multiplied to reach a single digit, Read munchausen Whether n equals the sum of its digits each raised to itself, taking 0^0 as 1. 1 and 3435 do. 0, 2, and 438579 Read munchausen_sum The sum of the digits of n each raised to itself, taking 0^0 as 1. 0 gives 1, 1 gives 1, 2 gives 4, 10 gives 2 Read music_chord Spell a chord up from a root: major, minor, diminished, augmented, sus2, sus4, dom7, maj7, min7, dim7, half-di Read music_interval The interval between two written notes. The number comes from the letters and the quality from the semitones, Read music_transpose Move a written note by a named interval. The letter moves with the interval number, then the accidental is cho Read mx_check MX / mail-routing check: sorted MX with inferred provider (Google, M365, Proofpoint, etc.), A/AAAA, SPF presen Read narayana The Narayana number N(n,k). N(5,3) is 20. A k outside 1..n yields 0. n runs from 1 to 60 and k from 0 to 60. P Read narayana_check The Narayana product C(n,k)C(n,k-1). It is exactly n times N(n,k). product is that binomial product and value Read narayana_cows Narayana's cow sequence at n, counting from 1. The first three terms are 1, and each later term is the previou Read narayana_row Narayana numbers N(n,1) through N(n,n). Their sum is the Catalan number C(n). The row for n = 4 is 1, 6, 6, 1 Read narcissistic Whether n equals the sum of its digits each raised to the number of digits. 153, 370, 371, and 407 do, and so Read neg_pell Whether x^2 - dy^2 = -1 has a positive integer solution. It does exactly when the continued-fraction period of Read neg_pell_count How many nonsquares from 1 through n have an odd continued-fraction period, so the negative Pell equation is s Read negabinary The base -2 digits of n, without a leading zero except for 0 itself. 0 is "0", 1 is "1", 2 is "110", 4 is "100 Read negabinary_bits How many 1 digits the base -2 writing of n has. 0 has none, 1 has 1, 6 has 3, and 7 has 4. n runs from 0 to 99 Read negabinary_decode The integer written by a base -2 bit string. "0" is 0, "110" is 2, "100" is 4, and "11010" is 6. A character o Read negabinary_digit The base -2 digit of n in place, counting from 0 at the right. For 6, which is "11010", place 0 is 0, place 1 Read negabinary_len How many digits the base -2 writing of n uses. 0 uses 1, 4 uses 3, and 6 uses 5. n runs from 0 to 999999999999 Read next_prime The least prime greater than or equal to n. 0, 1, and 2 all give 2. 14 gives 17. 10000000 gives 10000019. n ru Read nft_metadata ERC-721 / ERC-1155 token metadata: tokenURI/uri (ipfs/arweave/data/https), ownerOf, collection name/symbol/sup Read npv_irr Investment appraisal for a cash-flow series: NPV at a discount rate, IRR (all roots for non-conventional flows Read number_to_words Spell out numbers in words — cardinal, ordinal, currency amounts, cheque wording, years and Roman numerals — i Read octahedral The nth octahedral number, n(2n^2+1)/3. The first terms are 1, 6, 19, 44, and 85. n runs from 1 to 100000. Pri Read octahedral_index The n with O(n) equal to the input, or 0 when the input is not octahedral. 1 is place 1, 6 is place 2, 19 is p Read octahedral_mod The nth octahedral number modulo m. O(4) mod 10 is 4, O(5) mod 7 is 1, and O(6) mod 10 is 6. n runs from 1 to Read octahedral_next The smallest octahedral number that is at least n. 1 stays 1, 2 gives 6, 7 gives 19, and 45 gives 85. n runs f Read octahedral_sum The sum of the first n octahedral numbers. Through 1 it is 1, through 3 it is 26, through 6 it is 301, and thr Read odd_goldbach The lexicographically smallest triple of primes adding to an odd n. 7 is 2+2+3, 9 is 2+2+5, and 27 is 2+2+23. Read odd_square_sum The sum of the first n odd squares: 1^2 + 3^2 + ... + (2n-1)^2 = n(4n^2-1)/3. One term is 1, two terms are 10, Read ohm_law Ohm's law for one passive resistor. Give any two of voltage (volts), current (amperes), resistance (ohms) and Read openalex_search Scholarly search over OpenAlex (250M+ works): works with title, authors + ORCID + institutions, year, venue, D Read openapi_summary Summarise an OpenAPI 3.x or Swagger 2.0 API from a URL (SSRF-safe) or an inline JSON / YAML spec, so an agent Read openfda_drug FDA drug facts by brand/generic name or NDC from openFDA: label sections (boxed warning, indications, dosage, Read opengraph Extract Open Graph, Twitter Card, title/description, canonical, icons and JSON-LD @types from a public URL. Id Read ore Whether n is an Ore number: n times the divisor count is divisible by sigma(n), so the harmonic mean of the di Read osv_scan Known vulnerabilities for ONE package version from OSV.dev (GitHub Advisory DB, PyPA, Go, RustSec, OSS-Fuzz, d Read osv_scan_batch Dependency vulnerability scan (SCA) via OSV.dev querybatch: send up to 500 packages, or paste a lockfile (pack Read overpartition The number of overpartitions of n. The first part may be overlined. The values begin 1, 2, 4, 8, 14, 24, so n= Read padic The p-adic valuation v_p(n): how many times the prime p divides n. v_2(12) is 2, v_3(12) is 1, and v_5(12) is Read padovan The nth Padovan number. P0, P1, and P2 are 1, and each later term is the sum of the term two back and the term Read padovan_step The Padovan step identity P(n) = P(n-1) + P(n-5). left is P(n-1) and right is P(n-5). n runs from 5 to 8192. P Read padovan_sum The sum of Padovan numbers, or Perrin numbers, from lo through hi inclusive. seq is p or r. lo must not pass h Read palindrome Whether n reads the same forwards and backwards. 0, 7, and 121 do. 10 and 89 do not. n runs from 0 to 99999999 Read paper_fit How many copies of one sheet fit on another at full size. The copy may be turned. Two A4 pages fill an A3 and Read paper_next The next larger and smaller sheet in the same ISO series. A4 sits between A3 and A5. A0 has no larger size and Read parking Whether the preference list is a parking function. (1,1), (2,1), and (1,2,1) are. (2,2) and (1,3) are not. The Read parking_moved How many cars miss their preferred spot. (1,1) moves 1 car. (2,1) moves none. (1,2,1) moves 1. A list that is Read parking_shift The total of parked spot minus preference. (1,1) shifts by 1. (2,1) shifts by 0. (1,2,1) shifts by 2. A list t Read parking_spots Where each car parks, comma-separated. (1,1) parks at 1,2. (2,1) parks at 2,1. (1,2,1) parks at 1,2,3. A list Read parking_witness The first 1-based place where the sorted preferences exceed the place, or 0 when the list is a parking functio Read partition The number of integer partitions p(n), from the pentagonal recurrence. p(0) is 1, p(5) is 7, and p(10) is 42. Read partition_conjugate The conjugate of an integer partition, read down the columns of its Ferrers diagram. Parts must be positive an Read partition_distinct The number of partitions of n into distinct positive parts. This equals the number of partitions into odd part Read partition_parts Check an integer partition and return its sum and length. Parts must be positive and nonincreasing. The empty Read password_hash Hash, verify and identify password hashes — for seeding users, migrating auth systems, writing htpasswd files Read password_strength Offline password strength audit (POST only, so the password never appears in a URL or access log; never echoed Read pdi Whether n equals the sum of its digits each raised to k. 153 with k=3, 1634 with k=4, and 4150 with k=5 do. 41 Read pdi_count How many integers from 0 through n equal the sum of their digits each raised to k. For k=3, through 1 there ar Read pell The nth Pell number. P0 is 0, P1 is 1, and each later term is twice the previous one plus the one before that. Read pell_compose Compose two solutions of x^2 - 2y^2 = ±1. The unit for index a multiplied by the unit for index b is the unit Read pell_equation The nth solution of the Pell equation x^2 - 2y^2 = (-1)^n. x is half the companion Pell-Lucas number and y is Read pell_lucas The nth companion Pell-Lucas number. Q0 and Q1 are both 2, and later terms use the Pell recurrence. An optiona Read pell_sum The sum of Pell numbers, or companion Pell-Lucas numbers, from lo through hi inclusive. seq is p or q. lo must Read pentagonal The generalized pentagonal number n(3n-1)/2. n may be negative. n = 1 gives 1, n = -1 gives 2, and n = 0 is re Read pentanacci The nth Pentanacci number. P0 through P3 are 0, P4 is 1, and each later term is the sum of the previous five. Read pentanacci_index The smallest index n with P(n) equal to value. 0 is index 0, and 1 is index 4. A number that is not a Pentanac Read pentanacci_mod P(n) mod m for the Pentanacci sequence. The modulus is a positive decimal integer, and zero is rejected. n run Read pentanacci_step The Pentanacci step P(n) = P(n-1) + P(n-2) + P(n-3) + P(n-4) + P(n-5). a is P(n-5), b is P(n-4), c is P(n-3), Read pentanacci_sum The sum of Pentanacci numbers from lo through hi inclusive. The sum from P0 through P12 is 244. lo must not pa Read perfect_check Classify n from its divisor sum. 6 and 28 are perfect. 12 is abundant. 8 is deficient. The class is perfect, a Read perfect_power Whether n is a perfect power a^k with k at least 2. The base is the smallest such a, so the exponent is the la Read perfect_totient Whether n equals the sum of its iterated totients down to 1. The sum includes the final 1 and excludes n. 3, 9 Read period_double The period-doubling bit at index n, starting at 0. It is the parity of the exponent of 2 in n+1. Bit 0 is 0, b Read period_double_prefix The first n period-doubling bits, as a string of the digits 0 and 1. The first 8 bits are 01000101. n runs fro Read period_double_sum The number of 1s in the first n period-doubling bits, from index 0 through n-1. The first 8 bits have three 1s Read perm_compose Compose two permutations of the same length. a is applied first, then b, so the result at i is b[a[i]]. Both m Read perm_inverse Inverse of a permutation of 0..n-1. The inverse of [1, 2, 0] is [2, 0, 1]. Applying both is the identity. Pric Read perm_next The next permutation in lexicographic order. After [0, 1, 2] comes [0, 2, 1]. The reverse, [2, 1, 0], is the l Read perm_rank Lexicographic rank of a permutation, or the permutation at a rank. [0, 1, 2] is rank 0 and [2, 0, 1] is rank 4 Read perm_sign Sign of a permutation of 0..n-1, from the number of inversions. [2, 1, 0] has three inversions and sign -1. Th Read permission_check Whether one class may read, write, or execute under a mode. Pass the class that applies: on a real system the Read perrin The nth Perrin number. R0 is 3, R1 is 0, R2 is 2, and later terms use the Padovan recurrence. An optional modu Read perrin_divide The remainder of the nth Perrin number divided by n. A prime n divides R(n), so the remainder is 0. n runs fro Read pgp_key_info Inspect an OpenPGP public key (ASCII-armored or base64 binary) the way gpg --show-keys does, offline: primary Read phishing_score Heuristic phishing-likelihood score for a URL/domain: brand typosquat distance, subdomain depth, punycode, RDA Read pierpont Whether n is 2^a 3^b + 1. 2, 4, 10, 19, and 145 are. 1, 8, and 11 are not. n runs from 1 to 999999999999999. Read pierpont_count How many Pierpont primes are at most n. Through 1 there are 0, through 10 there are 4, through 20 there are 7, Read pierpont_next The smallest Pierpont prime that is at least n. 1 gives 2, 8 gives 13, 20 gives 37, and 98 gives 109. n runs f Read pierpont_powers The pair a,b with n = 2^a 3^b + 1. 2 gives 0,0. 10 gives 0,2. 19 gives 1,2. 145 gives 4,2. A number not of th Read pierpont_prime Whether n is a Pierpont prime, a prime of the form 2^a 3^b + 1. 2, 7, 19, and 109 are. 4, 10, 11, and 145 are Read pii_detect Detect and redact PII in text: emails, phone numbers (E.164, NANP, keyword-anchored national), credit cards (L Read ping Paid x402 test route for agents and client developers: the cheapest way to prove your x402 client signs a Sola Read plane_partition The number of plane partitions of n. The values begin 1, 1, 3, 6, 13, 24, so n=5 is 24, n=9 is 282, and n=10 i Read plane_partition_sum The sum of the plane partition numbers from 0 through n. Through 0 it is 1, through 2 it is 5, through 4 it is Read polite Whether n is polite: it is a sum of two or more consecutive positive integers. Every number that is not a powe Read polite_split The shortest sum of consecutive positive integers equal to n. 9 is 4+5, so it starts at 4 and has length 2. 6 Read poly_deriv Differentiate an integer polynomial. The derivative of a constant is the zero polynomial, which is an empty li Read poly_div Exact quotient of two integer polynomials. The divisor must be non-zero, and every leading step must divide in Read poly_eval Evaluate an integer polynomial at an integer point, by Horner. The zero polynomial evaluates to 0. An optional Read poly_mul Multiply two integer polynomials, lowest degree first. The zero polynomial is an empty list. An optional modul Read polydivisible Whether the first k digits of n form a multiple of k, for every k. 0, 10, 12, 24, 102, and 3816547290 do. 11 a Read polydivisible_count How many polydivisible numbers from 0 through n there are. Through 1 there are 2, through 10 there are 11, thr Read polygonal The nth s-gonal number, n((s-2)n - (s-4))/2. Triangular 4 is 10, square 5 is 25, pentagonal 5 is 35, and hexag Read polygonal_index The index n of an s-gonal number. Triangular 10 is index 4, and square 25 is index 5. 0 is index 0. A value th Read popcount The number of 1 bits in n. 0 has none, 255 has 8, and 256 has 1. n runs from 0 to 999999999999999. Price: $0.0 Read portfolio_metrics Portfolio and asset risk/return analytics from price or return series you supply: annualised return (CAGR) and Read position_size Risk-based position sizing for trading agents: from account balance, risk per trade (percent or fixed amount), Read power_mod a to the power e, reduced modulo m. 2^10 mod 1000 is 24, 3^5 mod 7 is 5, and (-2)^3 mod 7 is 6. A zero exponen Read powerful Whether n is powerful: 1, or every prime factor's exponent is at least 2. 1, 8, and 36 are powerful. 12 is not Read practical Whether n is practical: every integer from 1 to n is a sum of distinct divisors of n. 1, 2, 6, and 12 are prac Read practical_count How many practical numbers are from 1 through n. There are 5 through 10, and 15 through 40. n runs from 1 to 5 Read prev_prime The greatest prime less than or equal to n. 2 gives 2, 10 gives 7, and 10000000 gives 9999991. n runs from 2 t Read prime_check Whether n is prime, by trial division. 1 is not prime. n < 1 is rejected. n runs up to 10^12. Price: $0.001 US Read prime_count pi(n), the number of primes less than or equal to n. pi(0) and pi(1) are 0, pi(10) is 4, pi(100) is 25, and pi Read prime_gap The prime at or below n, the next prime after that one, and the difference. At 10 the prime is 7, the next is Read prime_nth The kth prime. The 1st is 2, the 10th is 29, the 100th is 541, and the 664579th is 9999991. k runs from 1 to 6 Read prime_power If n is a prime power p^k with k at least 1, return that prime and the exponent. Otherwise the prime and the e Read prime_triplet Whether n starts a prime triplet: n, n+2, and n+6 are prime, or n, n+4, and n+6 are prime. 5, 7, 11, and 13 do Read prime_triplet_count How many prime-triplet starters are at most n. Through 3 there are none, through 5 there is 1, through 13 ther Read primitive_abundant Whether n is abundant and no proper divisor is abundant. 12, 18, 20, and 70 are. 24 is not, because 12 divides Read primitive_root Whether g is a primitive root modulo the prime p. 3 is a primitive root mod 7, and 2 is not. 1 is a primitive Read primorial The primorial of n, the product of the primes less than or equal to n. The primorial of 1 is 1, of 10 is 210, Read primorial_prime Whether n + 1 and n - 1 are prime, where n is the product of the primes up to n. 5 is 30, and both neighbors a Read prompt_injection_scan Offline prompt-injection / jailbreak / exfiltration scanner: 10 rule families (ignore-instructions, reveal-sys Read pronic The nth pronic number, n(n+1). The 0th is 0, the 1st is 2, and the 5th is 30. n runs from 0 to 10000000. Price Read proth The Proth number k2^n + 1. k must be odd and strictly less than 2^n. 12^1+1 is 3, and 32^5+1 is 97. An even k, Read prufer_degree The degree of a vertex in the tree described by a Prüfer code. Degree is one plus the number of times the vert Read pubmed_search Search PubMed biomedical literature via NCBI E-utilities (esearch + esummary + efetch): PMID, title, authors, Read pythag_check Check a^2 + b^2 = c^2 for positive integers, with c the hypotenuse. primitive is true when the three sides are Read quadratic_residue Whether a is a square modulo an odd prime p, counting 0 as a square. 2 is a square mod 7, and 3 is not. -1 is Read quat_inv Inverse of a quaternion, the conjugate divided by the squared length. (1, 2, 3, 4) inverts to (1/30, -1/15, -1 Read quat_mul Hamilton product of two quaternions, scalar first. i times j is k, and j times i is minus k. Components are ex Read quat_norm Conjugate and squared length of a quaternion. (1, 2, 3, 4) has squared length 30 and conjugate (1, -2, -3, -4) Read quat_pow Integer power of a quaternion. i squared is -1, and i to the fourth is 1. Exponent 0 is 1, including zero to t Read quat_rotate Rotate a vector by a quaternion, q v q inverse. k sends (1, 0, 0) to (-1, 0, 0), a half turn about z. Scaling Read radical The radical of n, the product of its distinct prime factors. rad(1) is 1, rad(12) is 6, and rad(36) is 6. n ru Read range_lcm The least common multiple of the integers from lo through hi. The LCM of 1 through 10 is 2520, and the LCM of Read rdap_age Lightweight RDAP registration age lookup: registered/expiry dates, ageDays, daysToExpiry, registrar, status fl Read readability_extract Extract the main article from a web page (Readability-style content scoring, no headless browser): clean body Read recaman The nth Recamán term. a(0) is 0. Each later term steps back by n when that value is positive and new, and othe Read recaman_seen Whether value appears in Recamán's sequence from a(0) through a(limit), and the smallest index if it does. 2 a Read redirect_chain Follow HTTP redirects (max 10) with SSRF protections; report each hop status, Location, Server and Content-Typ Read refactorable Whether n is refactorable: the number of positive divisors of n divides n. 1, 2, 8, 9, and 12 are refactorable Read regex_explain Explain, lint and test a regular expression before an agent ships it. Parses the pattern (PCRE, JavaScript, Py Read regex_test Offline PCRE2 regex tester (JS-compatible i/m/s/u flags): match (with offsets + capture groups), replace, or s Read rencontres The rencontres number D(n, k), the permutations of n objects with k fixed points. D(n, k) = C(n, k) !(n-k). D Read repo_posture Security posture of a public GitHub repository from the OpenSSF Scorecard API: aggregate 0-10 score and rating Read repunit The repunit of length n, (10^n - 1)/9. R1 is 1 and R2 is 11. n = 0 is rejected. n runs up to 500. Past 15 digi Read resistor_network Series or parallel equivalent of 2 to 24 resistances. Series adds them. Parallel is the reciprocal of the sum Read reverse_dns Reverse DNS (PTR) for up to 20 IPv4 / IPv6 addresses with forward-confirmed reverse DNS (FCrDNS: does the PTR Read riordan The nth Riordan number. R(0) is 1, R(1) is 0, and (n+1)R(n) = (n-1)(2R(n-1) + 3R(n-2)) for n at least 2. R(4) Read riordan_mod R(n) mod m for the Riordan numbers. The modulus is a positive decimal integer, and zero is rejected. n runs fr Read riordan_sum The sum of Riordan numbers from lo through hi inclusive. The sum from R(0) through R(10) is 989. lo must not p Read rising The rising factorial n^(k) = n(n+1)...(n+k-1). 5^(3) is 210, and 5^(0) is 1. n runs from 0 to 100000 and k run Read robots_sitemap Fetch and parse robots.txt plus discovered sitemaps for a domain: user-agents, allow/disallow samples, sitemap Read rrule_expand Expand an iCalendar RFC 5545 RRULE (recurring event rule, as used by Google Calendar, Outlook, CalDAV) into co Read rudin_shapiro The Rudin-Shapiro bit of n, the parity of the number of overlapping 11 blocks in its binary expansion. n = 3 i Read ruler The ruler value of n, the exponent of 2 in n. v2(1) is 0, v2(8) is 3, and v2(12) is 2. n runs from 1 to 100000 Read ruler_max The largest ruler value on 1 through n, and the smallest index where it occurs. That index is the greatest pow Read ruler_prefix The ruler values v2(1) through v2(n), written as comma-separated integers. The first 8 values are 0,1,0,2,0,1, Read ruler_sum The sum of the ruler values v2(1) through v2(n). It equals n minus the number of 1-bits in n. The sum through Read ruth_aaron Whether n and n+1 have the same sum of prime factors, counted with repetition. 5, 8, 15, and 714 do. 6 and 9 d Read safeprime Whether n and (n-1)/2 are both prime. 5 comes from 2, 7 comes from 3, and 23 comes from 11. 3 and 17 are not s Read saml_decode Decode and inspect SAML 2.0 messages for SSO debugging: paste a SAMLResponse / SAMLRequest (base64 from an HTT Read sanctions_screen Sanctions screening of a person or company NAME against the official consolidated lists: US OFAC SDN + non-SDN Read scam_text_score Scam / fraud likelihood for messages, DMs, emails, posts and listings: 16 pattern families (crypto doubling gi Read schema_org_extract Extract all schema.org structured data from a public page (SSRF-safe fetch) or HTML: JSON-LD (including @graph Read schema_validate Validate JSON data against a JSON Schema (Ajv + formats). Body: { "schema": {...}, "data": {...} }. Returns va Read schroder The nth large Schröder number. S(0) is 1, S(1) is 2, and S(5) is 394. n runs from 0 to 64. Past 15 digits the Read schroder_little The nth little Schröder number. It is 1 at n = 0 and half the large Schröder number for n > 0. s(5) is 197. n Read sec_13f Institutional holdings from a fund manager's latest SEC 13F-HR: the information table parsed from EDGAR, aggre Read sec_edgar_search SEC EDGAR filings search: latest 10-K/10-Q/8-K (or any form) for a company by name, ticker or CIK, with direct Read sec_financials Normalized financial statements for any SEC filer by ticker or CIK, straight from EDGAR XBRL: revenue, gross p Read sec_form4 Insider trading from SEC Form 4: the latest Form 4 filings for a ticker or CIK parsed from EDGAR ownershipDocu Read secret_scan Scan code, config, logs, diffs or prompts for leaked credentials before you commit, share or send them to an L Read security_txt Fetch and validate a site's security.txt (RFC 9116) for vulnerability disclosure: tries /.well-known/security. Read self_count How many self numbers are at most n. Through 1 there is 1, through 10 there are 5, through 20 there are 6, and Read self_next The smallest self number that is at least n. 1 stays 1, 2 gives 3, 10 gives 20, and 21 gives 31. n runs from 1 Read self_nth The kth self number. The 1st is 1, the 5th is 9, the 10th is 64, and the 20th is 165. k runs from 1 to 2000. P Read self_number Whether n is a self number: no m has m plus its digit sum equal to n. 1, 20, and 108 are. 2, 10, and 21 are no Read self_seed m plus the sum of its decimal digits. 1 gives 2, 5 gives 10, 19 gives 29, and 99 gives 117. A self number is a Read semiperfect Whether some subset of the proper divisors of n sums to n. 6 and 12 are semiperfect. 10 and 70 are not. n runs Read semver Semantic versioning toolkit with npm (node-semver) semantics: validate and parse a version, compare two versio Read sentiment Sentiment analysis with VADER (Hutto & Gilbert 2014; MIT), the lexicon-and-rules model tuned for social media, Read seo_audit On-page SEO audit of one URL with a 0-100 score, letter grade and prioritised issues (high/medium/low): title Read seventh_sum The sum of the first n seventh powers, n^2(n+1)^2(3n^4+6n^3-n^2-4n+2)/24. The sum through 10 is 18080425. n ru Read sexy_prime Whether n and n+6 are both prime. 5 pairs with 11, 7 pairs with 13, and 3 does not pair with 9. n runs from 1 Read sigma_k The sum of the kth powers of the divisors of n. sigma_2(1) is 1, sigma_2(6) is 50, and sigma_2(12) is 210. n r Read sixth_sum The sum of the first n sixth powers, n(n+1)(2n+1)(3n^4+6n^3-3n+1)/42. The sum through 10 is 1978405. n runs fr Read sl2_det The determinant ad-bc of a 2 by 2 integer matrix [a, b, c, d]. The determinant of [1, 1, 0, 1] is 1, and the d Read sl2_inv The inverse of a 2 by 2 integer matrix whose determinant is 1 or -1. The inverse of [1, 1, 0, 1] is [1, -1, 0, Read sl2_mul The product of two integer 2 by 2 matrices, each written as [a, b, c, d]. [1, 1, 0, 1] times [1, 0, 1, 1] is [ Read sl2_pow The non-negative integer power of a 2 by 2 integer matrix. The exponent 0 is the identity [1, 0, 0, 1]. [1, 1, Read sl2_trace The trace a+d of a 2 by 2 integer matrix [a, b, c, d]. The trace of [1, 1, 0, 1] is 2, and the trace of [8, 5, Read slaac_address IPv6 address a host builds for itself from a /64 prefix and a MAC (SLAAC, RFC 4862). The interface id is the m Read smallest_prime_factor The smallest prime factor of n. It is 1 for 1, 2 for 12, 5 for 35, and 7 for 49. n runs from 1 to 100000000000 Read smarandache The smallest positive k such that n divides k!. 1 gives 1, 6 gives 3, 8 gives 4, 9 gives 6, 12 gives 4, and 16 Read smarandache_gap n minus S(n). 1 and 4 give 0, 6 gives 3, 8 gives 4, 10 gives 5, and 16 gives 10. n runs from 1 to 100000. Pric Read smarandache_least The smallest positive integer m with S(m) = k. k=1 gives 1, k=6 gives 9, k=8 gives 32, k=9 gives 27, and k=11 Read smarandache_quot S(n)! divided by n. 1 gives 1, 6 gives 1, 8 gives 3, 9 gives 80, 10 gives 12, and 16 gives 45. n runs from 1 t Read smarandache_sum The sum of S(k) from 1 through n. Through 1 it is 1, through 6 it is 18, through 10 it is 40, and through 12 i Read smith Whether n is a composite Smith number: its digit sum equals the digit sum of its prime factors with multiplici Read sociable Whether n lies on an aliquot cycle of length at least 3. 12496 does. 6, 220, and 25 do not. A chain past 30 st Read sociable_chain The aliquot cycle containing n, starting at n, comma-separated. 220 gives 220,284. 6 gives 6. 12496 gives 1249 Read sociable_length The length of the aliquot cycle containing n, or 0 when n is not on a cycle. 6 gives 1, 220 gives 2, 12496 giv Read sociable_min The smallest member of the aliquot cycle containing n. 284 gives 220 and 14288 gives 12496. A number that is n Read sociable_sum The sum of the aliquot cycle containing n. 6 gives 6, 220 gives 504, and 12496 gives 71056. A number that is n Read sol_account Inspect any Solana account: exists or not, SOL balance (lamports), owner program with a readable name, executa Read sol_balance SOL and SPL token balances (classic SPL Token + Token-2022) for a Solana wallet address, via public Solana RPC Read sol_network Solana mainnet status in one call: current slot and block height, epoch with progress % and estimated time to Read sol_pda Derive Solana program-derived addresses offline: findProgramAddress (canonical bump search) for any program an Read sol_priority_fees Solana priority-fee estimator from getRecentPrioritizationFees (last ~150 slots), globally or for the accounts Read sol_signatures Recent transaction signatures that touched a Solana address (wallet, token account, mint or program), newest f Read sol_token_meta Solana SPL / Token-2022 mint metadata: decimals, total supply, mint & freeze authority, Token-2022 extensions, Read sol_token_risk Heuristic risk screen for a Solana token mint: mint/freeze authority, Token-2022 extensions (permanent delegat Read sol_tx_status Fast confirmation check for up to 100 Solana transaction signatures in one call (getSignatureStatuses with his Read solana_tx Decode a Solana mainnet transaction via public RPC: status, fee, compute units, parsed instructions, inner ins Read solicited_node IPv6 solicited-node multicast address, ff02::1:ffXX:XXXX, from the low 24 bits of an IPv6 address or from the Read sopf The sum of the distinct prime factors of n. 1 gives 0, 8 gives 2, 12 gives 5, and 30 gives 10. A prime gives i Read sopfr The sum of prime factors of n with multiplicity. sopfr(1) is 0, sopfr(12) is 7, and sopfr(36) is 10. n runs fr Read sophie_germain Whether n and 2n+1 are both prime. 2 gives the safeprime 5, 11 gives 23, and 7 gives 15, which is not prime. n Read sphenic Whether n is a product of three distinct primes. 30, 42, 66, and 70 are. 12 and 60 are not, because of the squ Read sqrt_cf The continued fraction of sqrt(d): the integer part a0 and the repeating block, comma-separated. sqrt(2) is 1 Read sqrt_period The period of the continued fraction of sqrt(d). sqrt(2) has period 1, sqrt(3) has period 2, sqrt(7) has perio Read sqrt_tail The last term of the repeating block of sqrt(d), which is twice the integer part. sqrt(2) ends in 2, sqrt(3) e Read square_free_part n divided by its largest square factor. 12 gives 3, 18 gives 2, 36 gives 1, and 45 gives 5. n runs from 1 to 1 Read square_part The largest square dividing n. 1 gives 1, 8 gives 4, 12 gives 4, and 18 gives 9. n runs from 1 to 100000000000 Read square_sum The sum of the first n squares, n(n+1)(2n+1)/6. The sum through 1 is 1, and through 10 is 385. n runs from 0 t Read squarefree Whether n is squarefree, together with its radical. 1 and 30 are squarefree. 4 and 12 are not. n runs from 1 t Read squarefree_count How many square-free positive integers are at most n. Through 1 there is 1, through 4 there are 3, through 10 Read squarefree_part n divided by its largest square divisor. 12 gives 3, 8 gives 2, 18 gives 2, 36 gives 1, and a square-free n gi Read ssh_key_info Inspect SSH public keys and OpenSSH certificates from authorized_keys, known_hosts, .pub, RFC 4716 or PEM inpu Read ssl_blacklist abuse.ch SSL Blacklist (SSLBL): check whether a TLS certificate SHA1 fingerprint is blacklisted as malware / b Read ssl_cert_info Inspect a live TLS certificate: subject, issuer, SANs, validity window, days to expiry, SHA-256 fingerprint, k Read star_number The nth star number, 6n(n-1) + 1. The 1st is 1, the 2nd is 13, and the 5th is 121. n runs from 1 to 1000000. P Read stars_bars The number of ways to place n identical items into k distinct bins, which is C(n+k-1, n). Three items in two b Read stats_describe Descriptive statistics for one or many numeric series: count / missing, mean, median, mode, range, sample and Read stern Stern's diatomic sequence. s(0) is 0, s(1) is 1, even terms repeat s(n/2), and an odd term is the sum of its t Read stirling1 The signed Stirling number of the first kind s(n, k). s(4, 2) is 11 and s(5, 2) is -50. s(n, 0) is 0 for n > 0 Read stirling1_rec One step of s(n, k) = s(n-1, k-1) - (n-1)s(n-1, k). For n = 5 and k = 2, the previous term is -6, the scaled t Read stirling1_row Row n of the signed Stirling numbers of the first kind, from s(n, 0) through s(n, n). The absolute values sum Read stirling1_unsigned The unsigned Stirling number of the first kind, the absolute value of s(n, k). It counts permutations of n wit Read stirling2_row Row n of Stirling numbers of the second kind, from S(n,0) through S(n,n), plus the Bell number that sums the r Read string_case Convert identifiers and phrases between naming conventions: camelCase, PascalCase, snake_case, SCREAMING_SNAKE Read subfactorial The subfactorial !n, the number of derangements of n objects. !0 is 1, !1 is 0, !2 is 1, and !5 is 44. n runs Read subtraction_grundy The Grundy value of a heap of n in the subtraction game with the given move sizes. With moves 1, 2, and 3, the Read subtraction_row Grundy values from heap 0 through heap n in a subtraction game, comma-separated. With moves 1, 2, and 3, the r Read subtraction_win Whether the player to move wins a subtraction game on a heap of n with perfect play. A win is a nonzero Grundy Read summarize_extractive Extractive summary with TextRank (Mihalcea & Tarau 2004): sentences are split (abbreviation / initial / decima Read sun_times Sunrise, sunset, solar noon, day length, civil / nautical / astronomical twilight, golden hour and blue hour f Read superabundant Whether sigma(n)/n is strictly larger than sigma(m)/m for every m below n. The ratio is compared as sigma(n)m Read superfactorial The superfactorial sf(n), the product of k! for k from 1 through n. sf(1) is 1, sf(3) is 12, and sf(4) is 288. Read sylvester The nth Sylvester number. s(0) is 2, and each later term is s(s-1)+1. A negative index is rejected. n runs up Read tangent_number The nth tangent number. T1 is 1, T2 is 2, T3 is 16, and T4 is 272. It is read from the Bernoulli number B(2n). Read taxicab Whether n has at least two different writings as a sum of two positive cubes. 1729 and 4104 do. 91 and 100 do Read taxicab_count How many positive integers through n have at least two writings as a sum of two positive cubes. Through 100 th Read tech_stack Lightweight technology fingerprint from public HTML/headers: CMS, frameworks (Next/Nuxt/React/Vue), CDNs, anal Read technical_indicators Technical analysis on a price series you supply: SMA, EMA, RSI (Wilder), MACD (line, signal, histogram), Bolli Read ternary_digit The balanced ternary digit of n in place, counting from 0 at the right, as -1, 0, or 1. For 5, which is "1TT", Read ternary_len How many balanced ternary digits n uses. 0 uses 1, 5 uses 3, and 14 uses 4. n runs from 0 to 999999999999999. Read ternary_nonzero How many nonzero balanced ternary digits n uses. 0 has none, 3 has 1, 6 has 2, and 5 has 3. n runs from 0 to 9 Read ternary_value The integer written by balanced ternary digits, where T means -1. "0" is 0, "T" is -1, "1T" is 2, "10" is 3, a Read tetrahedral The nth tetrahedral number, n(n+1)(n+2)/6. The 5th is 35 and the 8th is 120. n runs from 0 to 1000000. Past 15 Read tetranacci The nth Tetranacci number. T0, T1, and T2 are 0, T3 is 1, and each later term is the sum of the previous four. Read tetranacci_index The smallest index n with T(n) equal to value. 0 is index 0, and 1 is index 3. A number that is not a Tetranac Read tetranacci_mod T(n) mod m for the Tetranacci sequence. The modulus is a positive decimal integer, and zero is rejected. n run Read tetranacci_step The Tetranacci step T(n) = T(n-1) + T(n-2) + T(n-3) + T(n-4). a is T(n-4), b is T(n-3), c is T(n-2), and d is Read tetranacci_sum The sum of Tetranacci numbers from lo through hi inclusive. The sum from T0 through T10 is 116. lo must not pa Read text_block Wrap a Helvetica block and give its height. Line breaking is the same as text-wrap. Leading defaults to 1.2 ti Read text_clean Clean and normalise messy text before embedding, indexing or prompting: repairs mojibake (UTF-8 read as Window Read text_ellipsis Cut one Helvetica line so it fits a width, ending with an ellipsis (…) when the text is too long. The ellipsis Read text_stats Text statistics: characters, words, unique words, sentences, paragraphs, reading/speaking time, Flesch readabi Read text_width Width of one line in Helvetica or Helvetica-Bold, in points. Glyph widths are the Adobe standard-14 AFM number Read text_wrap Wrap text to a measure in Helvetica or Helvetica-Bold. Hard breaks are LF, CR, or CRLF. Inside a paragraph, sp Read thabit The nth Thabit number, 32^n - 1. n=0 gives 2, n=1 gives 5, and n=2 gives 11. n runs from 0 to 256. Past 15 dig Read three_squares Whether n is a sum of three integer squares, and one nonnegative witness x <= y <= z when it is. It is impossi Read thue_cube Find a cube in a 0/1 string, a block xxx. 000 is a cube at index 0 of length 1. A Thue-Morse prefix has no cub Read thue_morse The nth Thue-Morse bit, the parity of the number of 1-bits in n. t(0) is 0, t(1) is 1, and t(2) is 1. n runs f Read thue_prefix The first n Thue-Morse bits as a string of 0 and 1. The first 8 bits are 01101001. n runs from 1 to 4096. Pric Read thue_sum The number of 1-bits in the Thue-Morse sequence from lo through hi inclusive. The sum from t(0) through t(7) i Read thue_weight The number of 1s in t(0) through t(n), and the bit t(n) itself. Through n = 7 the weight is 4. n runs from 0 t Read ticker_resolver Resolve a stock ticker (or CIK) to the SEC CIK, registrant name, exchange and sibling share classes from SEC E Read token_units Exact (string-math, no floating point) conversion between human token amounts and on-chain base units: wei / g Read totient Euler totient phi(n), the count of integers from 1 to n that are coprime to n. phi(1) is 1. n = 0 is rejected. Read totient_sum The sum of Euler totients from 1 through n. Through 1 it is 1, through 5 it is 10, and through 10 it is 32. n Read treasury_rates US Treasury rates: the daily par yield curve (1M-30Y, from treasury.gov) with day-over-day change in bp, 2s10s Read triangle_shape Classify a strict integer triangle. sides is equilateral, isosceles, or scalene. angles is acute, right, or ob Read trib_index The smallest index n with T(n) equal to value. 0 is index 0, and 1 is index 2. A number that is not a Tribonac Read trib_mod T(n) mod m. The modulus is a positive decimal integer, and zero is rejected. n runs from 0 to 4096. The residu Read trib_step The Tribonacci step T(n) = T(n-1) + T(n-2) + T(n-3). left is T(n-3), mid is T(n-2), and right is T(n-1). n run Read trib_sum The sum of Tribonacci numbers from lo through hi inclusive. The sum from T0 through T10 is 177. lo must not pa Read tribonacci The nth Tribonacci number. T0 and T1 are 0, T2 is 1, and each later term is the sum of the previous three. An Read trimorphic Whether n^3 ends with n. 1, 4, 5, 6, 9, 25, 49, 76, 125, and 999 do. 10 does not. n runs from 1 to 10000. Pric Read trinomial The central trinomial coefficient T(n), the coefficient of x^n in (1+x+x^2)^n. T(0) and T(1) are 1, T(2) is 3, Read trinomial_mod T(n) modulo m. T(5) mod 10 is 1, T(6) mod 100 is 41, and T(4) mod 5 is 4. n runs from 0 to 30 and m runs from Read trinomial_prime Whether T(n) is prime. T(2), T(3), and T(4) are prime. T(0), T(1), T(5), and T(6) are not. n runs from 0 to 30 Read trinomial_sum The sum of the central trinomial coefficients T(0) through T(n). Through 0 it is 1, through 1 it is 2, through Read trinomial_term The summand C(n,2k)C(2k,k) of the central trinomial coefficient. For n=5 the summands are 1, 20, and 30. A k p Read triperfect Whether sigma(n) = 3n. 120 and 672 are triperfect. 6 and 28 are perfect, so they are not. n runs from 1 to 100 Read twin_prime Whether n and n+2 are both prime. 3 pairs with 5, 5 pairs with 7, and 7 does not pair with 9. n runs from 1 to Read two_cubes How many ways n is a sum of two positive cubes, ignoring order and allowing the two cubes to be equal. 2 has 1 Read two_squares How many integer pairs (x, y), counting order and signs, satisfy x^2 + y^2 = n. There is 1 for 0, 4 for 1, and Read two_squares_pair Nonnegative integers x and y with x <= y and x^2 + y^2 = n, choosing the greatest y. 5 is 1 and 2. 25 is 0 and Read umask_mode Mode of a file or directory created under a umask. A file starts from 666 and a directory from 777, then the u Read unit_economics SaaS / subscription unit economics: customer lifetime value (margin-adjusted), LTV:CAC ratio, CAC payback mont Read unit_sum The reduced sum of unit fractions 1/d. Dens must be positive. An empty list or a non-positive denominator is r Read unitary_count How many unitary divisors n has. The count is 2 to the number of distinct prime factors, and 1 has one unitary Read unitary_sigma The sum of the unitary divisors of n. A unitary divisor shares no prime factor with the complementary divisor. Read url_links_extract Extract every link on a page as absolute URLs (RFC 3986 resolution, <base href> aware), classified internal/ex Read url_markdown Fetch a public HTML page and convert main/article content to Markdown (Turndown). Scripts/styles stripped; out Read us_macro US macro dashboard from BLS in one call: CPI and core CPI, unemployment, nonfarm payrolls, hourly earnings, pa Read validate_batch Offline format validator for up to 100 values: email, URL, IPv4/IPv6, domain, UUID (version/variant), credit c Read vampire Whether n is a vampire number. 1260, 1395, 1435, and 6880 are. 1000 and 1261 are not. n runs from 1 to 9999. P Read vampire_count How many vampire numbers are at most n. Through 1259 there are none, through 1500 there are 3, through 2187 th Read vampire_fangs The smaller-first fang pair of a vampire number, comma-separated. 1260 gives "21,60" and 6880 gives "80,86". A Read vampire_low The smaller fang of a vampire number. 1260 gives 21, 1395 gives 15, and 6880 gives 80. A non-vampire is reject Read vampire_next The smallest vampire number that is at least n. 1000 and 1260 give 1260, 1261 gives 1395, and 2188 gives 6880. Read vin_decode Decode a 17-character Vehicle Identification Number: offline structure check (allowed characters, ISO 3779 che Read visible_points How many lattice points (x, y) with 1 <= x,y <= n are visible from the origin, meaning gcd(x, y) = 1. The 1-sq Read voltage_divider Unloaded voltage divider. r1 is the top resistor, from the supply to the output. r2 is the bottom resistor, fr Read von_staudt The von Staudt-Clausen integer. For even m at least 2, B(m) plus the sum of 1/p over primes p with p-1 dividin Read wagstaff The Wagstaff number (2^p + 1)/3 for an odd prime p. p=3 gives 3, p=5 gives 11, and p=7 gives 43. An even numbe Read wagstaff_prime Whether the Wagstaff number (2^p + 1)/3 is prime. p=3, 5, 7, and 31 are. p=29 is not. An even number or a comp Read wallet_risk_basic Basic heuristic risk screen for a Solana or EVM (Ethereum/Base/Arbitrum) wallet: activity & age, bundled OFAC- Read wayback Check Internet Archive Wayback Machine availability for a URL (closest snapshot timestamp + archived URL). Pub Read weather_global Worldwide weather forecast for any latitude/longitude from MET Norway Locationforecast 2.0 (the Norwegian Mete Read weather_nws US weather from the National Weather Service for a latitude/longitude: 7-day (or hourly) forecast periods with Read web_to_text Fetch a public web page and return its main content as clean plain text (boilerplate stripped, lists as bullet Read webhook_verify Verify (or create) webhook signatures for Stripe, GitHub, Slack, Shopify, Twilio, Standard Webhooks / Svix, Pa Read wedderburn The nth Wedderburn-Etherington number. a(1) is 1, and a(1) through a(6) are 1, 1, 1, 2, 3, 6. Even and odd ind Read wedderburn_index The smallest index n with Wedderburn-Etherington number a(n) equal to value. 1 is index 1, and 6 is index 6. A Read wedderburn_mod a(n) mod m for the Wedderburn-Etherington numbers. The modulus is a positive decimal integer, and zero is reje Read wedderburn_split The Wedderburn-Etherington split of n. cross is the sum of a(i)a(n-i) over the pairs below the middle, and mid Read wedderburn_sum The sum of Wedderburn-Etherington numbers from lo through hi inclusive. The sum from a(1) through a(6) is 14. Read weird Whether n is weird: abundant, but not semiperfect. 70 is weird, and so is 836. 12 is abundant but semiperfect, Read wikipedia Wikipedia in one call: search any of 340+ language editions (titles, descriptions, snippets, thumbnails) and g Read wilson_prime Whether p^2 divides (p-1)! + 1. Inside this cap, 5 and 13 do, and 2, 3, 7, 11, and 17 do not. A composite is r Read wilson_quotient The Wilson quotient ((p-1)! + 1) / p for a prime p. The quotient of 5 is 5, of 7 is 103, and of 13 is 36846277 Read woodall The nth Woodall number, n2^n - 1. The 1st is 1, the 2nd is 7, and the 3rd is 23. n runs from 1 to 256. Past 15 Read wythoff The nth lower Wythoff number, floor(nphi), where phi is (1+sqrt(5))/2. A(1) is 1, A(2) is 3, and A(5) is 8. n Read wythoff_inverse Which Wythoff sequence contains m, and at which index. 1 is the lower number at index 1. 2 is the upper number Read wythoff_kind Which Wythoff sequence contains m, and the index. 1 is lower index 1. 2 is upper index 1. 4 is lower index 3. Read wythoff_lower The lower Wythoff number floor(nphi), phi the golden ratio. n=1 gives 1, n=2 gives 3, n=5 gives 8, and n=10 gi Read wythoff_pair The nth Wythoff pair. lower is floor(nphi) and upper is lower+n. The pair for n = 1 is 1 and 2. The pair for n Read wythoff_sum The sum of the lower Wythoff numbers from index 1 through n. Through 1 it is 1, through 5 it is 22, and throug Read wythoff_upper The nth upper Wythoff number, floor(nphi^2), which is A(n)+n. B(1) is 2, B(2) is 5, and B(5) is 13. n runs fro Read x402_probe x402 preflight for AI agents: inspect any paid endpoint BEFORE paying. Fetches the unpaid 402, decodes PAYMENT Read x509_decode Decode PEM/DER X.509 certificates, chains and CSRs offline: subject/issuer DN, serial, validity and days left, Read zeckendorf_bits The Zeckendorf bit string of n, from the largest used Fibonacci index down to F2. The bit string of 1 is 1, of Read zeckendorf_sum Sum Fibonacci numbers by index, using F1 = 1 and F2 = 1. Indices must be at least 2 and no two may be consecut Read zuckerman Whether n has no zero digit and the product of its digits divides n. 1, 11, 12, 24, and 36 do. 10, 13, and 22 Read zuckerman_count How many Zuckerman numbers are at most n. Through 9 there are 9, through 12 there are 11, through 24 there are Read zumkeller Whether the divisors of n split into two sets with equal sum. 6, 12, 20, and 945 do. 1, 10, and 18 do not. n r Read zumkeller_count How many Zumkeller numbers are at most n. Through 1 there are none, through 6 there is 1, through 12 there are Read zumkeller_half Half the sum of the divisors of n. 6 gives 6, 10 gives 9, and 12 gives 14. An odd divisor-sum is rejected. n r Read zumkeller_odd Whether n is an odd Zumkeller number. 945 is. 1, 9, and 15 are not, and an even Zumkeller number such as 6 is
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FAQ

Questions about ScorpTech x402 API

Can an AI agent delete data through the ScorpTech x402 API MCP server? +

Yes. The ScorpTech x402 API server exposes 11 destructive tools including collatz_drop, dedupe_records, evm_selector_decode. These permanently remove resources with no undo. PolicyLayer blocks destructive tools by default so they never reach the upstream server.

How do I prevent bulk modifications through ScorpTech x402 API? +

The ScorpTech x402 API server has 58 write tools including barcode_generate, base_convert, bell. Set a rate limit in your policy -- for example, 10 calls per hour prevents an agent from making more than 10 modifications per hour. PolicyLayer enforces this at the gateway, before calls reach ScorpTech x402 API.

How many tools does the ScorpTech x402 API MCP server expose? +

1012 tools across 4 categories: Destructive, Execute, Read, Write. 870 are read-only. 142 can modify, create, or delete data.

How do I enforce a policy on ScorpTech x402 API? +

Register the ScorpTech x402 API MCP server in PolicyLayer, apply the suggested rules above (adjust the limits to your use case), and point your AI client at the PolicyLayer proxy URL instead of the server directly. Your agents keep the same tools; PolicyLayer evaluates every call against policy before it executes. Nothing to install, live in minutes.

Enforce policy on every ScorpTech x402 API tool call.

Deterministic rules across all 1012 ScorpTech x402 API tools. Per-identity grants. Full audit log. Live in minutes. Nothing to install.

ENFORCE POLICY ON SCORPTECH X402 API →

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1012 ScorpTech x402 API tools catalogued and risk-classified — across an index of 46,500+ MCP servers.

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