A056672 Number of unitary and squarefree divisors of n! Also, number of divisors of the special squarefree part of n!, A055773(n).

1, 2, 4, 2, 4, 2, 4, 4, 4, 2, 4, 4, 8, 4, 4, 4, 8, 8, 16, 16, 16, 8, 16, 16, 16, 8, 8, 8, 16, 16, 32, 32, 32, 16, 16, 16, 32, 16, 16, 16, 32, 32, 64, 64, 64, 32, 64, 64, 64, 64, 64, 64, 128, 128, 128, 128, 128, 64, 128, 128, 256, 128, 128, 128, 128, 128, 256, 256, 256, 256

Offset: 1

Author

Labos Elemer, Aug 10 2000

Keywords

• nonn
• easy

Comments

The divisor d=1 is counted here as being free of prime divisors and also unitary.

Examples

			n=11: 11! = 2*2*2*2*2*2*2*2*3*3*3*3*5*5*7*11, has 540 divisors, 32 are unitary and 32 are squarefree. Only 4 divisors, {1,7,11,77} have both properties, so a(11)=4.
		

Links

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

Crossrefs

Cf. A000005, A000142, A007913, A055229, A055231, A055773, A056171.

Programs

Mathematica

rad[n_] := Times @@ First /@ FactorInteger[n]; p[n_] := Denominator[n/rad[n]^2]; a[n_] := DivisorSigma[0, p[n!]]; Array[a, 70] (* Amiram Eldar, Sep 22 2019 *)

PARI

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

PARI

a(n) = 1 << (primepi(n) - primepi(n>>1)); \\ Kevin Ryde, Jun 03 2023

Formula

a(n) = A000005(A055231(n!)).

a(n) = A000005(A007913(n!)/A055229(n!)).

a(n) = A000005(A055773(n)).

a(n) = 2^A056171(n). - Kevin Ryde, Jun 03 2023