A090630 - OEIS

%I #26 Jun 09 2022 06:32:15

%S 1,1,1,1,8,8,144,144,576,5184,518400,518400,2073600,2073600,101606400,

%T 914457600,14631321600,14631321600,526727577600,526727577600,

%U 52672757760000,221225582592000,6373403688960000,6373403688960000,917770131210240000,22944253280256000000,3877578804363264000000

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

%C a(n) is a square for all n except n = 4, 5 and 21 (Wilke, 1981). - _Amiram Eldar_, Jun 09 2022

%H Charles R Greathouse IV, <a href="/A090630/b090630.txt">Table of n, a(n) for n = 0..500</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/PerfectPower.html">Perfect Powers</a>.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Factorial.html">Factorial</a>.

%H 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 <a href="/index/Fa#factorial">Index entries for sequences related to factorial numbers</a>.

%F a(n)= n!/A251753(n). - _Robert G. Wilson v_, Dec 08 2014

%p f:= proc(n)

%p   local F,  k, d,r,s;

%p   F:= ifactors(n!)[2];

%p   r:= 1;

%p   for k from 2 to F[1][2] do

%p     r:= max(r, mul(f[1]^(k*floor(f[2]/k)),f=F))

%p   od:

%p r

%p end proc:

%p 1,1,seq(f(n), n=2..100); # _Robert Israel_, Dec 08 2014

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

%o (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 Cf. A011776, A060818, A060828, A251753.

%K nonn

%O 0,5

%A _Reinhard Zumkeller_, Dec 13 2003

%E More terms from _T. D. Noe_, Oct 04 2004