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

0,3

Links

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

Formula

a(n) = A037445(n!).

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

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

Mathematica

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

Prog

(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<<sum(e.bit_count() for e in sum((Counter(factorint(i)) for i in range(2, n+1)), start=Counter()).values()) # Chai Wah Wu, Aug 22 2024

Crossrefs

Cf. A027423, A037445, A048656, A177329.

Sequence in context: A070336 A348433 A053026 * A119318 A082953 A009289

Adjacent sequences: A375632 A375633 A375634 * A375636 A375637 A375638

Keyword

nonn

Author

Amiram Eldar, Aug 22 2024

Status

approved