A056629 Number of unitary square divisors of 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

Unitary analog of A046951(n!) = A055993(n).

Links

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

Formula

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!)].

Example

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.

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 *)

Crossrefs

Cf. A055993, A048656, A056657, A000188, A008833, A055229, A046951, A055230, A055071, A001221.

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

Status

approved