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Square root of largest square dividing n!.
19

%I #35 Jul 26 2024 15:56:53

%S 1,1,1,2,2,12,12,24,72,720,720,1440,1440,10080,30240,120960,120960,

%T 725760,725760,7257600,7257600,79833600,79833600,958003200,4790016000,

%U 62270208000,186810624000,2615348736000,2615348736000,15692092416000

%N Square root of largest square dividing n!.

%H Charles R Greathouse IV, <a href="/A055772/b055772.txt">Table of n, a(n) for n = 1..500</a>

%F a(n) = A000188(n!) = sqrt(A008833(n!)) = sqrt(A055071(n)).

%F n! = a(n)^2*A055204(n) = a(n)^2*A007913(n!).

%F n! = (A000188(n!)^2)*A055229(n!)*A055231(n!).

%F log(a(n)) ~ n*log(n)/2. - _David Radcliffe_, Oct 17 2014

%e For n=6, 6! = 720 = 144*5 so a(6) = sqrt(144) = 12.

%p a:= proc(n)

%p local r,F,t;

%p r:= n!;

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

%p mul(t[1]^floor(t[2]/2),t=F)

%p end proc:

%p seq(a(n), n= 1 .. 100); # _Robert Israel_, Oct 19 2014

%t Table[Last[Select[Sqrt[#]&/@Divisors[n!],IntegerQ]],{n,30}] (* _Harvey P. Dale_, Oct 08 2012 *)

%t (Sqrt@Factorial@Range@30)/.Sqrt[_]->1 (* _Morgan L. Owens_, May 04 2016 *)

%t f[p_, e_] := p^Floor[e/2]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n!]; Array[a, 40] (* _Amiram Eldar_, Jul 26 2024 *)

%o (PARI) a(n)=core(n!,2)[2] \\ _Charles R Greathouse IV_, Apr 03 2012

%Y Cf. A000188, A008833, A007913, A055229, A055231, A055071, A055204, A055230.

%K nonn

%O 1,4

%A _Labos Elemer_, Jul 12 2000