This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A056629 #14 Aug 03 2024 07:09:52
%S A056629 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 A056629 64,64,64,32,64,64,64,64,128,64,64,64,64,64,128,128,256,256,256,256,
%U A056629 256,128,256,256,256,256,256,128,256,256,256,256,512,512,512,512,512
%N A056629 a(n) = A034444(A056627(n)).
%C A056629 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
%e A056629 a(10) = A034444(A056627(10)) = A034444(720) = 8.
%t A056629 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 *)
%t A056629 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 *)
%o A056629 (PARI) 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
%Y A056629 Cf. A008833, A034444, A055229, A056627, A375187.
%K A056629 nonn
%O A056629 1,6
%A A056629 _Labos Elemer_, Aug 08 2000
%E A056629 Incorrect name replaced with a formula by _Amiram Eldar_, Aug 03 2024