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A055772 Square root of largest square dividing n!.

%I A055772 #35 Jul 26 2024 15:56:53
%S A055772 1,1,1,2,2,12,12,24,72,720,720,1440,1440,10080,30240,120960,120960,
%T A055772 725760,725760,7257600,7257600,79833600,79833600,958003200,4790016000,
%U A055772 62270208000,186810624000,2615348736000,2615348736000,15692092416000
%N A055772 Square root of largest square dividing n!.
%H A055772 Charles R Greathouse IV, <a href="/A055772/b055772.txt">Table of n, a(n) for n = 1..500</a>
%F A055772 a(n) = A000188(n!) = sqrt(A008833(n!)) = sqrt(A055071(n)).
%F A055772 n! = a(n)^2*A055204(n) = a(n)^2*A007913(n!).
%F A055772 n! = (A000188(n!)^2)*A055229(n!)*A055231(n!).
%F A055772 log(a(n)) ~ n*log(n)/2. - _David Radcliffe_, Oct 17 2014
%e A055772 For n=6, 6! = 720 = 144*5 so a(6) = sqrt(144) = 12.
%p A055772 a:= proc(n)
%p A055772 local r,F,t;
%p A055772 r:= n!;
%p A055772 F:= ifactors(r)[2];
%p A055772 mul(t[1]^floor(t[2]/2),t=F)
%p A055772 end proc:
%p A055772 seq(a(n), n= 1 .. 100); # _Robert Israel_, Oct 19 2014
%t A055772 Table[Last[Select[Sqrt[#]&/@Divisors[n!],IntegerQ]],{n,30}] (* _Harvey P. Dale_, Oct 08 2012 *)
%t A055772 (Sqrt@Factorial@Range@30)/.Sqrt[_]->1 (* _Morgan L. Owens_, May 04 2016 *)
%t A055772 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 A055772 (PARI) a(n)=core(n!,2)[2] \\ _Charles R Greathouse IV_, Apr 03 2012
%Y A055772 Cf. A000188, A008833, A007913, A055229, A055231, A055071, A055204, A055230.
%K A055772 nonn
%O A055772 1,4
%A A055772 _Labos Elemer_, Jul 12 2000