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A226460 Let m! have prime factorization Product (p_j^e_j); a(n) = number of distinct prime factors p_j such that e_j = n has no solution for any m!. 2
0, 0, 1, 1, 0, 2, 1, 2, 0, 1, 0, 3, 2, 2, 1, 1, 1, 3, 0, 1, 2, 1, 0, 4, 2, 1, 0, 2, 1, 4, 2, 2, 0, 2, 0, 2, 2, 3, 1, 3, 1, 2, 1, 3, 1, 1, 0, 5, 2, 0, 0, 2, 2, 2, 1, 3, 2, 0, 1, 5, 3, 3, 1, 1, 2, 2, 0, 2, 1, 3, 0, 4, 2, 3, 0, 2, 1, 2, 1, 4, 2, 0, 0, 6, 1, 1, 0 (list; graph; refs; listen; history; text; internal format)
OFFSET

0,6

COMMENTS

If n belongs to A048247 then a(n) is equal to zero.

For a given prime p and n satisfying p^k + p^(k-1) + ... + 1 <= n < p^(k+1) + ... + 1 for some k, let r_k = n mod (p^k + p^(k-1) + ... + 1), r_(k-1) = r_k mod (p^(k-1) + ... + 1), and so on down to r_1 = r_2 mod (p + 1). Then, p^n appears in a factorial m! iff none of the r_i is congruent to -1. - Charlie Neder, Nov 03 2018

LINKS

Jinyuan Wang, Table of n, a(n) for n = 0..1000

EXAMPLE

For n = 11, there are three distinct prime factors (3, 5, 11) in factorization of m!.

3^10 divides 26! ( 26! is not divisible by 3^11).

3^13 divides 27!.

5^10 divides 49! ( 49! is not divisible by 5^11).

5^12 divides 50!.

11^10 divides 120! ( 120! is not divisible by 11^11).

11^12 divides 121!.

The exponent of three distinct prime factors never becomes equal to 11. (It searches for all the exponent of prime factorization of factorials [A000142].)

Therefore a(11)=3.

PROG

(PARI) is(k, p) = my(c, s); while(s<k, c++; s+=1+valuation(c, p)); s>k;

a(n) = sum(p=2, n, isprime(p)&&is(n, p)); \\ Jinyuan Wang, Aug 22 2021

CROSSREFS

Cf. A000142, A048247, A115627.

Sequence in context: A340795 A137853 A213890 * A094114 A104607 A120728

Adjacent sequences:  A226457 A226458 A226459 * A226461 A226462 A226463

KEYWORD

nonn,easy

AUTHOR

Naohiro Nomoto, Jun 08 2013

STATUS

approved

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Last modified December 8 20:37 EST 2021. Contains 349596 sequences. (Running on oeis4.)