2e:null 2f:["$","title",null,{"children":"OEIS Sequence Implementations | ƒ | Observable"}] 34:[] 32:{"id":"3e2df326bcd91299","slug":"oeis","title":"OEIS Sequence Implementations","thumbnail":"8ec17eafe9905f8e1744a3cb99257cc9746ea7d0ce0afa474fd0e2c03d775759","access_level":"public","create_time":"$D2019-03-19T01:19:03.210Z","update_time":"$D2019-06-25T03:07:49.874Z","view_time":null,"first_public_time":"$D2019-03-19T04:44:45.541Z","has_unpublished_edits":null,"fork":false,"stars":0,"starred":false,"owner":{"type":"team","id":"01c5b3348f0ad41e","login":"follymath","name":"ƒ","avatar_url":"https://avatars-next.observableusercontent.com/6ab71fe2b6ec1b0072af6ff766fbf76def55b1e24d5463f2deb67e7274337e67","tier":"starter_2024"},"version":697,"publish_version":697,"role":null,"authored":false,"authors":[{"type":"individual","id":"e83e96b3489ba704","login":"follymath","name":"ƒ","avatar_url":"https://avatars-next.observableusercontent.com/6ab71fe2b6ec1b0072af6ff766fbf76def55b1e24d5463f2deb67e7274337e67","tier":null}],"members":[{"type":"individual","id":"e83e96b3489ba704","login":"follymath","name":"ƒ","avatar_url":"https://avatars-next.observableusercontent.com/6ab71fe2b6ec1b0072af6ff766fbf76def55b1e24d5463f2deb67e7274337e67","tier":null,"role":"editor"}],"comments":[],"custom_thumbnail":false,"trashed":false,"lapsed":null,"subscribed":null,"forks":0,"collection":null,"type":"notebook","body":{"title":"OEIS Sequence Implementations","cells":[{"id":0,"value":"md`\n# OEIS Sequence Implementations\n\nThis is an attempt at accumulating various OEIS implementations, along with a collection of common utilities\nhelpful for implementations. The observablehq import mechanisms ought to make this relatively easy to tease\napart into one notebook sequence as this list grows. We could also pull series metadata and values generically\nfrom oeis.org -- and do some interesting plotting, etc. (both of distinct series, and given enough series\nimplementations, of useful aggregate data a la [Sloane's gap](https://en.wikipedia.org/wiki/On-Line_Encyclopedia_of_Integer_Sequences#Sloane's_gap).\n`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":35,"value":"md`---\n## [A000002](http://oeis.org/A000002)\n\nKolakoski sequence: ${tex`a_n`} is length of ${tex`n^{th}`} run; ${tex`a_1 = 1`}; sequence consists just of 1's and 2's.\n\nIt is an unsolved problem to show that the density of 1's is equal to 1/2.\n\nA weaker problem is to construct a combinatorial bijection between the set of positions of 1's and the set of positions of 2's.\n`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":28,"value":"function* A000002(n=1) {\n let x = -1\n let y = x\n // TODO fast forward to index position n...\n const out = [2, 1]\n do {\n yield out[x&1]\n const f = y &~ (y+1)\n x ^= f\n y = (y+1) | (f & (x>>1))\n } while (++n < Number.MAX_SAFE_INTEGER)\n}","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":64,"value":"A000002_expected = [1,2,2,1,1,2,1,2,2,1,2,2,1,1,2,1,1,2,2,1,2,1,1,2,\n 1,2,2,1,1,2,1,1,2,1,2,2,1,2,2,1,1,2,1,2,2,1,2,1,1,\n 2,1,1,2,2,1,2,2,1,1,2,1,2,2,1,2,2,1,1,2,1,1,2,1,2,\n 2,1,2,1,1,2,2,1,2,2,1,1,2,1,2,2,1,2,2,1,1,2,1,1,2,\n 2,1,2,1,1,2,1,2,2]","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":76,"value":"assert(A000002_expected, take(A000002_expected.length, A000002()))","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":96,"value":"md`---\n## [A000004](http://oeis.org/A000004)\n\nThe zero sequence.\n`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":109,"value":"function* A000004(n=0) {\n do {\n yield 0\n } while (++n <= Number.MAX_SAFE_INTEGER)\n}","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":97,"value":"A000004_expected = [0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,\n 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,\n 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,\n 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,\n 0,0,0]","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":101,"value":"assert(A000004_expected, take(A000004_expected.length, A000004()))","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":120,"value":"md`---\n## [A000005](http://oeis.org/A000005)\n${tex`d(n)`} (also called ${tex`\\tau(n)`} or ${tex`\\sigma_0(n)`}), the number of divisors of ${tex`n`}.\n\nNumber of factors in the factorization of the polynomial ${tex`x^n-1`} over ${tex`\\mathbb{Z}`}.\n\n${tex`a_n = 2 \\iff n \\in \\mathbb{P}`}.\n`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":150,"value":"function* A000005(n=1) {\n // TODO quite a bit of room to optimize\n do {\n if (n==1) {\n yield 1\n continue\n }\n const pp = primePowers(n)\n let v = 1\n for (let [p, e] of pp) {\n v *= (e + 1)\n }\n yield v\n } while (++n <= Number.MAX_SAFE_INTEGER)\n}","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":152,"value":"A000005_expected = [1,2,2,3,2,4,2,4,3,4,2,6,2,4,4,5,2,6,2,6,4,4,2,8,\n 3,4,4,6,2,8,2,6,4,4,4,9,2,4,4,8,2,8,2,6,6,4,2,10,\n 3,6,4,6,2,8,4,8,4,4,2,12,2,4,6,7,4,8,2,6,4,8,2,12,\n 2,4,6,6,4,8,2,10,5,4,2,12,4,4,4,8,2,12,4,6,4,4,4,\n 12,2,6,6,9,2,8,2,8]","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":154,"value":"assert(A000005_expected, take(A000005_expected.length, A000005()))","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":322,"value":"md`---\n## [A000006](http://oeis.org/A000006)\n\nInteger part of the square root of ${tex`n^{th}`} prime.\n\nConjecture: No two successive terms in the sequence differ by more than 1. Proof of this would prove the converse of the theorem that every prime is surrounded by two consecutive squares, namely ${tex`\\lfloor{\\sqrt{p}}\\rfloor^2`} and ${tex`\\lfloor{\\sqrt{p} + 1}\\rfloor^2`}.\n\nEquals the number of squares less than prime(n).\n`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":335,"value":"function* A000006(n=1) {\n // TODO seek to index position n\n for (let p of primes()) {\n yield Math.floor(Math.sqrt(p))\n }\n}","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":328,"value":"A000006_expected = [1,1,2,2,3,3,4,4,4,5,5,6,6,6,6,7,7,7,8,8,8,8,9,9,\n 9,10,10,10,10,10,11,11,11,11,12,12,12,12,12,13,13,\n 13,13,13,14,14,14,14,15,15,15,15,15,15,16,16,16,\n 16,16,16,16,17,17,17,17,17,18,18,18,18,18]","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":330,"value":"assert(A000006_expected, take(A000006_expected.length, A000006()))","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":337,"value":"md`---\n## [A000007](http://oeis.org/A000007)\nThe characteristic function of 0: ${tex`a_n = 0^n`}.\n`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":359,"value":"function* A000007(n=0) {\n do {\n yield (n ? 0 : 1)\n } while (++n <= Number.MAX_SAFE_INTEGER)\n}","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":339,"value":"A000007_expected = [1,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,\n 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,\n 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,\n 0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,0,\n 0,0,0,0,0,0]","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":363,"value":"assert(A000007_expected, take(A000007_expected.length, A000007()))","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":366,"value":"md`---\n## [A000010](http://oeis.org/A000010)\nEuler totient function ${tex`\\varphi(n)`}: count numbers ≤ n and prime to n.\n`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":376,"value":"function* A000010(n=1) {\n do {\n if (n==1) {\n yield 1\n continue\n }\n let v = n\n // loop over distinct factors\n for (let p of new Set([...factors(n)])) {\n v *= p-1\n v /= p\n }\n yield v\n } while (++n <= Number.MAX_SAFE_INTEGER)\n}","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":371,"value":"A000010_expected = [1,1,2,2,4,2,6,4,6,4,10,4,12,6,8,8,16,6,18,8,12,\n 10,22,8,20,12,18,12,28,8,30,16,20,16,24,12,36,18,\n 24,16,40,12,42,20,24,22,46,16,42,20,32,24,52,18,\n 40,24,36,28,58,16,60,30,36,32,48,20,66,32,44]","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":373,"value":"assert(A000010_expected, take(A000010_expected.length, A000010()))","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":396,"value":"md`---\n## [A000012](http://oeis.org/A000012)\n\nThe simplest sequence of positive numbers: the all 1's sequence.\n`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":401,"value":"function* A000012(n=0) {\n // TODO why does OEIS have these indices running backward?\n do {\n yield 1\n } while (++n <= Number.MAX_SAFE_INTEGER)\n}","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":344,"value":"md`---\n## [A000027](http://oeis.org/A000027)\n\nThe positive integers, ${tex`\\mathbb{N^*}`}.\n`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":354,"value":"function* A000027(n=1) {\n do {\n yield n\n } while (++n <= Number.MAX_SAFE_INTEGER)\n}","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":350,"value":"A000027_expected = [1,2,3,4,5,6,7,8,9,10,11,12,13,14,15,16,17,18,19,\n 20,21,22,23,24,25,26,27,28,29,30,31,32,33,34,35,\n 36,37,38,39,40,41,42,43,44,45,46,47,48,49,50,51,\n 52,53,54,55,56,57,58,59,60,61,62,63,64,65,66,67,\n 68,69,70,71,72,73,74,75,76,77]","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":356,"value":"assert(A000027_expected, take(A000027_expected.length, A000027()))","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":404,"value":"md`---\n## [A000037](http://oeis.org/A000037)\n\nNumbers that are not squares.\n\n${tex`n \\mapsto n + \\lfloor{\\frac{1}{2} + \\sqrt{n}}\\rfloor`}\n`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":406,"value":"function* A000037(n=1) {\n // while (n <= Number.MAX_SAFE_INTEGER) ?\n do {\n yield n + Math.floor(1/2 + Math.sqrt(n))\n } while (++n <= Number.MAX_SAFE_INTEGER)\n}","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":423,"value":"A000037_expected = [2,3,5,6,7,8,10,11,12,13,14,15,17,18,19,20,21,22,\n 23,24,26,27,28,29,30,31,32,33,34,35,37,38,39,40,\n 41,42,43,44,45,46,47,48,50,51,52,53,54,55,56,57,\n 58,59,60,61,62,63,65,66,67,68,69,70,71,72,73,74,\n 75,76,77,78,79,80,82,83,84,85,86,87,88,89,90,91,\n 92,93,94,95,96,97,98,99]","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":425,"value":"assert(A000037_expected, take(A000037_expected.length, A000037()))","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":430,"value":"assert(take(1,A000037(1234))[0], 1269) // value at 1,234th index position of sequence","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":498,"value":"assert(A000037(1234).next().value, 1269)","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":11,"value":"md`---\n## [A000040](http://oeis.org/A000040)\n\nThe prime numbers.\n\nElements of ${tex`\\mathbb{P}`} ordered by magnitude.\n\nA natural number is prime if and only if it has exactly two (positive) divisors.\n\nA prime has exactly one proper positive divisor, 1.\n`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":4,"value":"A000040 = primes // TODO take an n","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":310,"value":"A000040_expected = [2,3,5,7,11,13,17,19,23,29,31,37,41,43,47,53,59,\n 61,67,71,73,79,83,89,97,101,103,107,109,113,127,\n 131,137,139,149,151,157,163,167,173,179,181,191,\n 193,197,199,211,223,227,229,233,239,241,251,257,\n 263,269,271]","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":16,"value":"assert(A000040_expected, [...itertools.takewhile(A000040(), (p) => p <= 271)])","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":459,"value":"md`---\n## [A000042](http://oeis.org/A000042)\n\nUnary representation of natural numbers.\n`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":464,"value":"function* A000042(n=1, b='') {\n // obviously this falls down fast without bigints...\n // could force strings here instead, though prob want a custom numeric type anyway\n // opting into custom type accomplished with second \"base\" arg\n do {\n if (typeof b === 'string') {\n yield Array.from({ length: n }).fill('1').join('')\n } else if (typeof b === 'number') {\n // base \"b\" repdigit\n // TODO BigInt polyfill?\n yield (b**n - 1) / (b - 1)\n }\n } while (++n <= Number.MAX_SAFE_INTEGER)\n}","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":452,"value":"A000042_expected = ['1','11','111','1111','11111','111111','1111111','11111111',\n '111111111','1111111111','11111111111','111111111111',\n '1111111111111','11111111111111','111111111111111',\n '1111111111111111','11111111111111111',\n '111111111111111111','1111111111111111111',\n '11111111111111111111']","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":467,"value":"assert(A000042_expected, take(A000042_expected.length, A000042()))","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":507,"value":"assert(A000042(5,10).next().value, 11111)","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":524,"value":"assert(A000042(5,2).next().value, parseInt('11111',2))","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":600,"value":"md`---\n## [A000217](http://oeis.org/A000217)\n\nTriangular numbers: ${tex`a_n = \\binom{n+1}{2} = \\frac{n(n+1)}{2} = 0 + 1 + 2 + \\ldots + n`}.\n\nReflection: ${tex`a_n = a_{-1-n}`}.\n\n${tex`a_n + a_{n-1} * a_{n+1} = a_n^2`}.\n`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":619,"value":"function* A000217(n=0) {\n // TODO support characters over an alphabet?\n do {\n yield choose(n+1,2)\n } while (++n <= Number.MAX_SAFE_INTEGER)\n}","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":621,"value":"A000217_expected = [0,1,3,6,10,15,21,28,36,45,55,66,78,91,105,120,\n 136,153,171,190,210,231,253,276,300,325,351,378,\n 406,435,465,496,528,561,595,630,666,703,741,780,\n 820,861,903,946,990,1035,1081,1128,1176,1225,1275,\n 1326,1378,1431]","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":615,"value":"assert(A000217_expected, take(A000217_expected.length, A000217()))","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":581,"value":"md`---\n## [A000290](http://oeis.org/A000290)\n\nThe squares: ${tex`a(n) = n^2`}.\n`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":585,"value":"function* A000290(n=0) {\n // TODO support characters over an alphabet?\n do {\n yield n*n\n } while (++n <= Number.MAX_SAFE_INTEGER)\n}","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":596,"value":"A000290_expected = [0,1,4,9,16,25,36,49,64,81,100,121,144,169,196,\n 225,256,289,324,361,400,441,484,529,576,625,676,\n 729,784,841,900,961,1024,1089,1156,1225,1296,1369,\n 1444,1521,1600,1681,1764,1849,1936,2025,2116,2209,\n 2304,2401,2500]","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":588,"value":"assert(A000290_expected, take(A000290_expected.length, A000290()))","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":536,"value":"md`---\n## [A006881](http://oeis.org/A006881)\n\nSquarefree semiprimes: Numbers that are the product of two distinct primes.\n\nNumbers ${tex`n`} such that ${tex`\\varphi(n) + \\sigma(n) = 2(n+1)`}.\n\n${tex`n`} such that ${tex`\\tau(n) = \\omega(n)^{\\omega(n)}`}.\n\n${tex`a_n \\approx \\frac{n\\log n}{\\log\\log n}`}.\n`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":556,"value":"function* A006881(n=1) {\n // seek to first value... TODO should we `yield null` for each place?\n if (n < 6) n = 6\n do {\n // TODO base generator to emit pairs of factors, can be composed back together\n if (n % 2 == 0) {\n // for singly even n, divide out by 2 and run a prime test\n if (n % 4 == 2 && isPrime(n/2)) yield n\n } else {\n // seek out first prime factor, test for primality of residue\n // TODO we can be smarter than this by doubly iterating primes\n const f1 = factors(n).next().value\n const f2 = n / f1\n if (f1 != f2 && isPrime(f2)) yield n\n }\n } while (++n <= Number.MAX_SAFE_INTEGER)\n}","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":531,"value":"A006881_expected = [6,10,14,15,21,22,26,33,34,35,38,39,46,51,55,57,\n 58,62,65,69,74,77,82,85,86,87,91,93,94,95,106,111,\n 115,118,119,122,123,129,133,134,141,142,143,145,\n 146,155,158,159,161,166,177,178,183,185,187,194,\n 201,202,203,205]","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":566,"value":"assert(A006881_expected, take(A006881_expected.length, A006881()))","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":168,"value":"md`---\n## Utilities\n`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":218,"value":"take = (n, iterable) => [...itertools.itake(n, itertools.icompact(iterable))]","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":166,"value":"function* factors(n) {\n let x = Math.max(1, Math.floor(n)) || 1\n let i = 0\n let q\n for (const p of primes()) {\n if (!(p * p <= x)) break;\n while ((q = x / p) > 1 && q === Math.floor(q)) {\n x = q\n yield p\n }\n if (++i % 1000 === 0) {\n yield\n }\n }\n yield x\n}","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":260,"value":"primePowers = (n) => countValues(itertools.groupby([...itertools.icompact(factors(n))]))","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":270,"value":"countValues = (grouped) => [...itertools.imap(grouped, ([k, v]) => [k, [...v].length])]","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":635,"value":"function choose(n, k) {\n if (k === 0) return 1\n return (n * choose(n-1, k-1)) / k\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":9,"value":"md`---\n## Dependencies`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":5,"value":"import {primes} from \"@mourner/fast-prime-generator\"","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":559,"value":"import {isPrime} from '@randomfractals/primes'","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":88,"value":"import {assert} from \"@mootari/mutable-forms\"","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":43,"value":"itertools = require('https://bundle.run/itertools@1.3.2')","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"}],"files":"$W34","stdlib":"1","titling":"auto"}} 33:"$32:body"