A056675 Number of non-unitary but squarefree divisors of n!. Also number of unitary but not-squarefree divisors of n!.

0, 0, 0, 2, 4, 6, 12, 12, 12, 14, 28, 28, 56, 60, 60, 60, 120, 120, 240, 240, 240, 248, 496, 496, 496, 504, 504, 504, 1008, 1008, 2016, 2016, 2016, 2032, 2032, 2032, 4064, 4080, 4080, 4080, 8160, 8160, 16320, 16320, 16320, 16352, 32704, 32704, 32704, 32704

Offset: 1

Author

Labos Elemer, Aug 10 2000

Keywords

• nonn

Examples

			n=10: 10! = 2*2*2*2*2*2*2*2*3*3*3*3*5*5*7 = 256*81*25*7, which has 270 divisors, of which 16 are unitary and 16 are squarefree; overlap={1,7}. The set {2, 3, 5, 6, 10, 14, 15, 21, 30, 35, 42, 70, 105, 210} represents the squarefree non-unitary divisors of 10!, so a(10)=14.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000

Crossrefs

Cf. A000005, A000142, A007913, A048656, A055229, A055231, A056674, A056772.

Programs

PARI

a(n) = my(f=n!); sumdiv(f, d, issquarefree(d) && (gcd(d, f/d) != 1)); \\ Michel Marcus, Sep 05 2017

Formula

a(n) = A048656(n) - A000005(A055231(n!)) = A048656(n) - A000005(A007913(n!)/A055229(n!)).

a(n) = A056674(A000142(n)). - Amiram Eldar, Aug 06 2019

a(n) = A048656(n) - A056672(n). - Sean A. Irvine, May 02 2022