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

%I A090630 #27 Feb 16 2025 08:32:52
%S A090630 1,1,1,1,8,8,144,144,576,5184,518400,518400,2073600,2073600,101606400,
%T A090630 914457600,14631321600,14631321600,526727577600,526727577600,
%U A090630 52672757760000,221225582592000,6373403688960000,6373403688960000,917770131210240000,22944253280256000000,3877578804363264000000
%N A090630 Greatest divisor d of n! such that d=m^k with k>1.
%C A090630 a(n) is a square for all n except n = 4, 5 and 21 (Wilke, 1981). - _Amiram Eldar_, Jun 09 2022
%H A090630 Charles R Greathouse IV, <a href="/A090630/b090630.txt">Table of n, a(n) for n = 0..500</a>
%H A090630 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/PerfectPower.html">Perfect Powers</a>.
%H A090630 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Factorial.html">Factorial</a>.
%H A090630 Kenneth M. Wilke, <a href="http://www.pme-math.org/journal/issues/PMEJ.Vol.7.No.4.pdf">Problem 493</a>, Pi Mu Epsilon Journal, Vol. 7, No. 4 (1981), p. 265; <a href="https://www.jstor.org/stable/24337449">alternative link</a>.
%H A090630 <a href="/index/Fa#factorial">Index entries for sequences related to factorial numbers</a>.
%F A090630 a(n)= n!/A251753(n). - _Robert G. Wilson v_, Dec 08 2014
%p A090630 f:= proc(n)
%p A090630   local F,  k, d,r,s;
%p A090630   F:= ifactors(n!)[2];
%p A090630   r:= 1;
%p A090630   for k from 2 to F[1][2] do
%p A090630     r:= max(r, mul(f[1]^(k*floor(f[2]/k)),f=F))
%p A090630   od:
%p A090630 r
%p A090630 end proc:
%p A090630 1,1,seq(f(n), n=2..100); # _Robert Israel_, Dec 08 2014
%t A090630 IsPower[n_] := If[n==1, True, GCD@@(Transpose[FactorInteger[n]][[2]])>1]; Table[Select[Divisors[n! ], IsPower][[ -1]], {n, 0, 25}]
%o A090630 (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
%Y A090630 Cf. A011776, A060818, A060828, A251753.
%K A090630 nonn
%O A090630 0,5
%A A090630 _Reinhard Zumkeller_, Dec 13 2003
%E A090630 More terms from _T. D. Noe_, Oct 04 2004