A090630 Greatest divisor d of n! such that d=m^k with k>1.

1, 1, 1, 1, 8, 8, 144, 144, 576, 5184, 518400, 518400, 2073600, 2073600, 101606400, 914457600, 14631321600, 14631321600, 526727577600, 526727577600, 52672757760000, 221225582592000, 6373403688960000, 6373403688960000, 917770131210240000, 22944253280256000000, 3877578804363264000000

Offset: 0

Author

Reinhard Zumkeller, Dec 13 2003

Keywords

• nonn

Comments

a(n) is a square for all n except n = 4, 5 and 21 (Wilke, 1981). - Amiram Eldar, Jun 09 2022

Links

• Charles R Greathouse IV, Table of n, a(n) for n = 0..500
• Eric Weisstein's World of Mathematics, Perfect Powers.
• Eric Weisstein's World of Mathematics, Factorial.
• Kenneth M. Wilke, Problem 493, Pi Mu Epsilon Journal, Vol. 7, No. 4 (1981), p. 265; alternative link.
• Index entries for sequences related to factorial numbers.

Crossrefs

Cf. A011776, A060818, A060828, A251753.

Programs

Maple

f:= proc(n)
  local F,  k, d,r,s;
  F:= ifactors(n!)[2];
  r:= 1;
  for k from 2 to F[1][2] do
    r:= max(r, mul(f[1]^(k*floor(f[2]/k)),f=F))
  od:
r
end proc:
1,1,seq(f(n), n=2..100); # Robert Israel, Dec 08 2014

Mathematica

IsPower[n_] := If[n==1, True, GCD@@(Transpose[FactorInteger[n]][[2]])>1]; Table[Select[Divisors[n! ], IsPower][[ -1]], {n, 0, 25}]

PARI

a(n)=my(f=factor(n!),m=1); for(i=2,if(#f~,f[1,2]), m=max(factorback(concat(Mat(f[,1]), f[,2]\i*i)),m)); m \\ Charles R Greathouse IV, Dec 09 2014

Formula

a(n)= n!/A251753(n). - Robert G. Wilson v, Dec 08 2014

Extensions

More terms from T. D. Noe, Oct 04 2004