A055772 Square root of largest square dividing n!.

1, 1, 1, 2, 2, 12, 12, 24, 72, 720, 720, 1440, 1440, 10080, 30240, 120960, 120960, 725760, 725760, 7257600, 7257600, 79833600, 79833600, 958003200, 4790016000, 62270208000, 186810624000, 2615348736000, 2615348736000, 15692092416000

Offset: 1

Author

Labos Elemer, Jul 12 2000

Keywords

• nonn

Examples

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

Links

• Charles R Greathouse IV, Table of n, a(n) for n = 1..500

Crossrefs

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

Programs

Maple

a:= proc(n)
local r,F,t;
r:= n!;
F:= ifactors(r)[2];
mul(t[1]^floor(t[2]/2),t=F)
end proc:
seq(a(n), n= 1 .. 100); # Robert Israel, Oct 19 2014

Mathematica

Table[Last[Select[Sqrt[#]&/@Divisors[n!],IntegerQ]],{n,30}] (* Harvey P. Dale, Oct 08 2012 *)
(Sqrt@Factorial@Range@30)/.Sqrt[]->1 (* _Morgan L. Owens, May 04 2016 *)
f[p_, e_] := p^Floor[e/2]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n!]; Array[a, 40] (* Amiram Eldar, Jul 26 2024 *)

PARI

a(n)=core(n!,2)[2] \\ Charles R Greathouse IV, Apr 03 2012

Formula

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

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

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

log(a(n)) ~ n*log(n)/2. - David Radcliffe, Oct 17 2014