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a(n) = A055993(n) - A034444(A056627(n)).
0

%I #14 Aug 03 2024 07:09:45

%S 0,0,0,1,1,2,2,4,8,22,22,28,28,56,88,120,120,172,172,284,292,584,584,

%T 848,1136,2272,2656,4304,4304,5312,5312,6080,6112,12992,16256,19376,

%U 19376,38752,43136,47936,47936,63936,63936,100672,132928,278528,278528

%N a(n) = A055993(n) - A034444(A056627(n)).

%C 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

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

%t 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 *)

%t 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 *)

%o (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

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

%K nonn

%O 1,6

%A _Labos Elemer_, Aug 08 2000

%E Incorrect name replaced with a formula by _Amiram Eldar_, Aug 03 2024