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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. 2
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 (list; graph; refs; listen; history; text; internal format)
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 A053626 A333696

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

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Last modified September 30 23:44 EDT 2020. Contains 337440 sequences. (Running on oeis4.)