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

1,4

Links

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

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

Example

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.

Prog

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

Crossrefs

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

Sequence in context: A180304 A332824 A053146 * A062857 A061799 A076868

Adjacent sequences: A056672 A056673 A056674 * A056676 A056677 A056678

Keyword

nonn

Author

Labos Elemer, Aug 10 2000

Status

approved