

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
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OFFSET

1,2


COMMENTS

Old name was: "If the prime factorization of n is n=product{pn} 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)+P1).


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 5^225, 2^126, 3^227, and 7^128.


PROG

(PARI) isokp(m, perm, f, nb) = {for (k=1, nb, if ((m+k1) % 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



