A162961 a(n) is the smallest k such that there exists a permutation b_0, b_1, ... of the maximal prime powers dividing n satisfying b_i | k+i for all i.

1, 2, 3, 4, 5, 2, 7, 8, 9, 4, 11, 3, 13, 6, 5, 16, 17, 8, 19, 4, 6, 10, 23, 8, 25, 12, 27, 7, 29, 3, 31, 32, 11, 16, 14, 8, 37, 18, 12, 15, 41, 6, 43, 11, 9, 22, 47, 15, 49, 24, 17, 12, 53, 26, 10, 7, 18, 28, 59, 3, 61, 30, 27, 64, 25, 9, 67, 16, 23, 5, 71, 8

Offset

1,2

Comments

Old name was: "If the prime factorization of n is n=product{p|n} p^b(n,p), then a(n) = the smallest positive integer such that each p^b(n,p) divides a different integer k where a(n) <= k <= a(n)+{number distinct prime divisors of n}-1. a(1)=1."

Clarification of definition: Each b(n,p) is a positive integer. Let the number of distinct primes dividing n be P. There is a permutation (c(1),c(2),c(3),...c(P)) of the p^b(n,p)'s such that c(1)|a(n), c(2)|(a(n)+1), c(3)|(a(n)+2),..., c(P)|(a(n)+P-1).

Links

Charlie Neder, Table of n, a(n) for n = 1..1000

Formula

a(n) = n iff n is a prime power (A000961). - Charlie Neder, Feb 03 2019

Example

3150 is factored as 2^1 * 3^2 * 5^2 * 7^1. a(3150) = 25, which can be seen from 5^2|25, 2^1|26, 3^2|27, and 7^1|28.

Prog

(PARI) isokp(m, perm, f, nb) = {for (k=1, nb, if ((m+k-1) % f[perm[k], 1]^f[perm[k], 2], return (0)); ); return (1); }
isokm(m, f, nb) = {for (j=0, nb!-1, my(perm = numtoperm(nb, j)); if (isokp(m, perm, f, nb), return (1)); ); return (0); }
a(n) = {my(f = factor(n), nb = #f~, m = 1); while (! isokm(m, f, nb), m++); m; } \\ Michel Marcus, Feb 05 2019

Crossrefs

Cf. A000961.

Sequence in context: A028233 A216972 A066296 * A145255 A344758 A053626

Adjacent sequences: A162958 A162959 A162960 * A162962 A162963 A162964

Keyword

nonn

Author

Leroy Quet, Jul 19 2009

Extensions

Name changed and a(31)-a(72) by Charlie Neder, Feb 03 2019

Status

approved