A056630 Number of non-unitary square divisors of n!.

0, 0, 0, 1, 1, 2, 2, 4, 8, 22, 22, 28, 28, 56, 88, 120, 120, 172, 172, 284, 292, 584, 584, 848, 1136, 2272, 2656, 4304, 4304, 5312, 5312, 6080, 6112, 12992, 16256, 19376, 19376, 38752, 43136, 47936, 47936, 63936, 63936, 100672, 132928, 278528, 278528

Offset

1,6

Links

Table of n, a(n) for n=1..47.

Formula

a(n) = A046951(n!) - 2^r, where r is the number of prime divisors of unitary analog of largest square divisor of n! and 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 22 is not unitary [but squares]: {4,9,16,36,64,100,144,225,324,400,576,...,57600,129600}. E.g. GCD[576,10!/576]=GCD[576,6300]=36. Thus a(10)=30-8=22.

Mathematica

A046951[n_] := Length[Select[Divisors[n], IntegerQ[Sqrt[#]] &]]; 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[A046951[n!] - 2^(PrimeNu[Sqrt[A008833[n!]]/A055229[n!]]), {n, 1, 50}] (* G. C. Greubel, May 20 2017 *)

Crossrefs

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

Sequence in context: A283117 A334325 A100396 * A259813 A121853 A064451

Adjacent sequences: A056627 A056628 A056629 * A056631 A056632 A056633

Keyword

nonn

Author

Labos Elemer, Aug 08 2000

Status

approved