A056672 - OEIS

%I #18 Jun 03 2023 23:56:33

%S 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,

%T 32,32,32,16,16,16,32,16,16,16,32,32,64,64,64,32,64,64,64,64,64,64,

%U 128,128,128,128,128,64,128,128,256,128,128,128,128,128,256,256,256,256

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

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

%H Amiram Eldar, <a href="/A056672/b056672.txt">Table of n, a(n) for n = 1..10000</a>

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

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

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

%F a(n) = 2^A056171(n). - _Kevin Ryde_, Jun 03 2023

%e 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.

%t 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 *)

%o (PARI) a(n) = my(f=n!); sumdiv(f, d, issquarefree(d) && (gcd(d, f/d) == 1)); \\ _Michel Marcus_, Sep 05 2017

%o (PARI) a(n) = 1 << (primepi(n) - primepi(n>>1)); \\ _Kevin Ryde_, Jun 03 2023

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

%K nonn,easy

%O 1,2

%A _Labos Elemer_, Aug 10 2000