A375635 The number of infinitary divisors of n!.

1, 1, 2, 4, 8, 16, 8, 16, 64, 64, 16, 32, 128, 256, 512, 1024, 2048, 4096, 256, 512, 512, 2048, 4096, 8192, 8192, 16384, 32768, 65536, 16384, 32768, 65536, 131072, 524288, 2097152, 131072, 65536, 32768, 65536, 131072, 262144, 524288, 1048576, 4194304, 8388608

Offset: 0

Author

Amiram Eldar, Aug 22 2024

Keywords

• nonn

Links

• Amiram Eldar, Table of n, a(n) for n = 0..3000

Crossrefs

Cf. A027423, A037445, A048656, A177329.

Programs

Mathematica

f[p_, e_] := 2^DigitCount[e, 2, 1]; a[0] = a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n!]; Array[a, 100, 0]

PARI

a(n) = vecprod(apply(x -> 2^hammingweight(x), factor(n!)[,2]));

Python

from collections import Counter
from sympy import factorint
def A375635(n): return 1<Chai Wah Wu, Aug 22 2024

Formula

a(n) = A037445(n!).

a(n) = 2^A177329(n).

A048656(n) <= a(n) <= A027423(n).