A375188 Number of non-unitary square divisors of n!.

0, 0, 0, 0, 1, 1, 2, 2, 6, 10, 22, 22, 32, 32, 68, 92, 124, 124, 172, 172, 284, 296, 596, 596, 848, 1136, 2288, 2680, 4352, 4352, 5344, 5344, 6128, 6140, 13040, 16304, 19424, 19424, 38864, 43184, 47984, 47984, 63992, 63992, 100784, 133024, 278656, 278656, 331520

Offset: 0

Author

Amiram Eldar, Aug 03 2024

Keywords

• nonn
• easy

Links

• Amiram Eldar, Table of n, a(n) for n = 0..1000

Crossrefs

Cf. A000142, A056626, A375187.

Programs

Mathematica

f1[p_, e_] := 1 + Floor[e/2]; f2[p_, e_] := 2^(1 - Mod[e, 2]); a[0] = a[1] = 0; a[n_] := Times @@ f1 @@@ (fct = FactorInteger[n!]) - Times @@ f2 @@@ fct; Array[a, 60, 0]

PARI

a(n) = {my(e = factor(n!)[,2]); vecprod(apply(x -> x\2 + 1, e)) - vecprod(apply(x -> 1 << (1 - x%2), e));}

Python

from math import prod
from collections import Counter
from sympy import factorint
def A375188(n):
    f = sum((Counter(factorint(m)) for m in range(2,n+1)),start=Counter()).values()
    return prod((e>>1)+1 for e in f)-(1<Chai Wah Wu, Aug 04 2024

Formula

a(n) = A056626(n!).