A056629 a(n) = A034444(A056627(n)).

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, 64, 64, 64, 32, 64, 64, 64, 64, 128, 64, 64, 64, 64, 64, 128, 128, 256, 256, 256, 256, 256, 128, 256, 256, 256, 256, 256, 128, 256, 256, 256, 256, 512, 512, 512, 512, 512

Offset

1,6

Comments

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

Links

Table of n, a(n) for n=1..66.

Example

a(10) = A034444(A056627(10)) = A034444(720) = 8.

Mathematica

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 *)
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 *)

Prog

(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

Crossrefs

Cf. A008833, A034444, A055229, A056627, A375187.

Sequence in context: A047210 A360915 A120327 * A245356 A167185 A081676

Adjacent sequences: A056626 A056627 A056628 * A056630 A056631 A056632

Keyword

nonn

Author

Labos Elemer, Aug 08 2000

Extensions

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

Status

approved