A375188 -id:A375188 - cp's OEIS frontend

A375187 Number of unitary square divisors of n!.

1, 1, 1, 1, 1, 1, 4, 4, 2, 2, 8, 8, 4, 4, 4, 4, 4, 4, 8, 8, 16, 4, 4, 4, 16, 16, 16, 8, 16, 16, 32, 32, 16, 4, 16, 16, 16, 16, 16, 16, 16, 16, 8, 8, 16, 32, 128, 128, 256, 256, 128, 32, 64, 64, 256, 64, 16, 4, 16, 16, 64, 64, 64, 128, 128, 32, 64, 64, 128, 128

Offset: 0

Amiram Eldar, Aug 03 2024

Unitary analog of A046951(n!) = A055993(n).

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

Cf. A000142, A046951, A055993, A056624, A375188.

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

a(n) = vecprod(apply(x -> 1 << (1 - x%2), factor(n!)[,2]));

from collections import Counter
from sympy import factorint
def A375187(n): return 1<Chai Wah Wu, Aug 03 2024

a(n) = A056624(n!).

A056630 a(n) = A055993(n) - A034444(A056627(n)).

Original entry on oeis.org

0, 0, 0, 1, 1, 2, 2, 4, 8, 22, 22, 28, 28, 56, 88, 120, 120, 172, 172, 284, 292, 584, 584, 848, 1136, 2272, 2656, 4304, 4304, 5312, 5312, 6080, 6112, 12992, 16256, 19376, 19376, 38752, 43136, 47936, 47936, 63936, 63936, 100672, 132928, 278528, 278528

Offset: 1

Labos Elemer, Aug 08 2000

nonn

Previous name, "Number of non-unitary square divisors of n!." was incorrect. See A375188 for the correct sequence with this name. - Amiram Eldar, Aug 03 2024

			example: a(10) = A055993(10) - A034444(A056627(10)) = 30 - A034444(720) = 30 - 8 = 22.

Cf. A008833, A034444, A046951, A055993, A055229, A056627, A375188.

A046951[n_] := Length[Select[Divisors[n], IntegerQ[Sqrt[#]] &]]; 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[A046951[n!] - 2^(PrimeNu[Sqrt[A008833[n!]]/A055229[n!]]), {n,1,50}] (* G. C. Greubel, May 20 2017 *)
f1[p_, e_] := 1 + Floor[e/2]; f2[p_, 1] := 1; f2[p_, e_] := If[EvenQ[e], p^(e/2), p^((e-3)/2)]; ; a[1] = 0; a[n_] := Times @@ f1 @@@ (fct = FactorInteger[n!]) - 2^PrimeNu[Times @@ f2 @@@ fct]; Array[a, 60] (* Amiram Eldar, Aug 03 2024 *)

a(n) = {my(f = factor(n!)); prod(i = 1, #f~, 1 + f[i, 2]\2) - 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

A375187 Number of unitary square divisors of n!.

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