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

0,7

Links

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

Formula

a(n) = A056626(n!).

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]

Prog

(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<<sum(e&1^1 for e in f)) # Chai Wah Wu, Aug 04 2024

Crossrefs

Cf. A000142, A056626, A375187.

Sequence in context: A167399 A247326 A374297 * A300274 A019310 A078008

Adjacent sequences: A375185 A375186 A375187 * A375189 A375190 A375191

Keyword

nonn,easy

Author

Amiram Eldar, Aug 03 2024

Status

approved