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A056629 Number of unitary square divisors of n!. 0
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 (list; graph; refs; listen; history; text; internal format)
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: A240835 A047210 A120327 * A245356 A167185 A081676

Adjacent sequences:  A056626 A056627 A056628 * A056630 A056631 A056632

KEYWORD

nonn

AUTHOR

Labos Elemer, Aug 08 2000

STATUS

approved

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Last modified September 25 21:33 EDT 2017. Contains 292500 sequences.