A375187 -id:A375187 - cp's OEIS frontend

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

Amiram Eldar, Aug 03 2024

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

Cf. A000142, A056626, A375187.

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]

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

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

a(n) = A056626(n!).

A056629 a(n) = A034444(A056627(n)).

Original entry on oeis.org

1, 1, 1, 1, 1, 4, 4, 4, 4, 8, 8, 8, 8, 16, 8, 8, 8, 8, 8, 16, 8, 16, 16, 16, 16, 32, 32, 64, 64, 64, 64, 64, 32, 64, 64, 64, 64, 128, 64, 64, 64, 64, 64, 128, 128, 256, 256, 256, 256, 256, 128, 256, 256, 256, 256, 256, 128, 256, 256, 256, 256, 512, 512, 512, 512, 512

Offset: 1

Labos Elemer, Aug 08 2000

nonn

Previous name, "Number of unitary square divisors of n!." was incorrect. See A375187 for the correct sequence with this name. - Amiram Eldar, Aug 03 2024

			a(10) = A034444(A056627(10)) = A034444(720) = 8.

Cf. A008833, A034444, A055229, A056627, A375187.

A008833[n_] := First[Select[Reverse[Divisors[n]], IntegerQ[Sqrt[#]] &, 1]]; A055229[n_] := With[{sf = Times @@ Power @@@ ({#[[1]], Mod[#[[2]], 2]} & /@ FactorInteger[n])}, GCD[sf, n/sf]]; Table[2^(PrimeNu[Sqrt[A008833[n!]]/A055229[n!]]), {n, 1, 50}] (* G. C. Greubel, May 19 2017 *)
f[p_, 1] := 1; f[p_, e_] := If[EvenQ[e], p^(e/2), p^((e-3)/2)]; a[1] = 1; a[n_] := 2^PrimeNu[Times @@ f @@@ FactorInteger[n!]]; Array[a, 66] (* Amiram Eldar, Aug 03 2024 *)

a(n) = {my(f = factor(n!)); 2^omega(prod(i = 1, #f~, if(f[i, 2] == 1, 1, f[i, 1]^if(f[i, 2]%2, (f[i, 2]-3)/2, f[i, 2]/2))));} \\ Amiram Eldar, Aug 03 2024

Incorrect name replaced with a formula by Amiram Eldar, Aug 03 2024

cp's OEIS Frontend

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

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