A117134 Greatest k such that n^k divides (n^2)!.

3, 4, 7, 6, 17, 8, 21, 20, 24, 12, 70, 14, 32, 55, 63, 18, 80, 20, 99, 73, 48, 24, 191, 78, 56, 121, 130, 30, 224, 32, 204, 108, 72, 203, 323, 38, 80, 126, 398, 42, 293, 44, 193, 505, 96, 48, 575, 200, 312, 162, 225, 54, 485, 302, 522, 180, 120, 60, 898, 62, 128, 660, 682

Offset: 2

Author

Robert Israel, Apr 26 2007

Keywords

• nonn
• look

Comments

If p is prime, a(p) = p+1, a(p^2) = floor((p^3 + p^2 + p + 1)/2).

Examples

			a(3)=4 because (3^2)! = 362880 = 3^4 * 4480 and 4480 is not divisible by 3.
		

References

• Thread "100!" in rec.puzzles newsgroup, April 2007

Links

• Robert Israel, Table of n, a(n) for n = 2..10000 (n=2..103 from Vincenzo Librandi)

Crossrefs

Cf. A011776.

Programs

Maple

seq(ordp((n^2)!,n), n=2..50);
# Alternative:
f:= proc(n) local F,m,t,v,j;
  F:= ifactors(n)[2];
  m:= infinity;
  for t in F do
    v:= add(floor(n^2/t[1]^j),j=1..ceil(log[t[1]](n^2)));
    m:= min(m,floor(v/t[2]));
  od;
  m
end proc:
map(f, [$2..100]); # Robert Israel, Feb 26 2019

Mathematica

gkn[n_]:=Module[{c=(n^2)!,k},k=Floor[Log[c]/Log[n]]; While[!Divisible[ c,n^k], k--];k]; Array[gkn,70,2] (* Harvey P. Dale, Sep 14 2012 *)