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%I #9 May 20 2017 21:55:08
%S 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,
%T 64,64,64,32,64,64,64,64,128,64,64,64,64,64,128,128,256,256,256,256,
%U 256,128,256,256,256,256,256,128,256,256,256,256,512,512,512,512,512
%N Number of unitary square divisors of n!.
%C Unitary analog of A046951(n!) = A055993(n).
%F a(n) = 2^r, where r is the number of prime divisors of unitary analog of largest square divisor of n!, where r(n) = A001221[A000188(n!) /A055229(n!)].
%e n=10 and the largest square-root divisor of 10! is 720. 10! has 30 square divisors, of which 8 is unitary [and squares]: {1,25,81,256,2025,6400,20736,518400}. E.g. GCD[256,10!/256]=GCD[256,14175]=1. Thus a(10)=8.
%t 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 *)
%Y Cf. A055993, A048656, A056657, A000188, A008833, A055229, A046951, A055230, A055071, A001221.
%K nonn
%O 1,6
%A _Labos Elemer_, Aug 08 2000
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