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

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

Author

Labos Elemer, Aug 08 2000

Keywords

• nonn

Comments

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

Examples

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

Crossrefs

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

Programs

Mathematica

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 *)

PARI

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

Extensions

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