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A056629
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Number of unitary square divisors of n!.
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0
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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
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OFFSET
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1,6
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COMMENTS
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Unitary analog of A046951(n!) = A055993(n).
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LINKS
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Table of n, a(n) for n=1..66.
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FORMULA
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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!)].
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EXAMPLE
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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.
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MATHEMATICA
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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 *)
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CROSSREFS
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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
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KEYWORD
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nonn
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AUTHOR
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Labos Elemer, Aug 08 2000
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STATUS
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approved
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