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

%I A375188 #14 Aug 04 2024 01:25:20
%S A375188 0,0,0,0,1,1,2,2,6,10,22,22,32,32,68,92,124,124,172,172,284,296,596,
%T A375188 596,848,1136,2288,2680,4352,4352,5344,5344,6128,6140,13040,16304,
%U A375188 19424,19424,38864,43184,47984,47984,63992,63992,100784,133024,278656,278656,331520
%N A375188 Number of non-unitary square divisors of n!.
%H A375188 Amiram Eldar, <a href="/A375188/b375188.txt">Table of n, a(n) for n = 0..1000</a>
%F A375188 a(n) = A056626(n!).
%t A375188 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]
%o A375188 (PARI) a(n) = {my(e = factor(n!)[,2]); vecprod(apply(x -> x\2 + 1, e)) - vecprod(apply(x -> 1 << (1 - x%2), e));}
%o A375188 (Python)
%o A375188 from math import prod
%o A375188 from collections import Counter
%o A375188 from sympy import factorint
%o A375188 def A375188(n):
%o A375188     f = sum((Counter(factorint(m)) for m in range(2,n+1)),start=Counter()).values()
%o A375188     return prod((e>>1)+1 for e in f)-(1<<sum(e&1^1 for e in f)) # _Chai Wah Wu_, Aug 04 2024
%Y A375188 Cf. A000142, A056626, A375187.
%K A375188 nonn,easy
%O A375188 0,7
%A A375188 _Amiram Eldar_, Aug 03 2024