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

1,4

Links

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

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

Example

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

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 *)

Prog

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

Crossrefs

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

Sequence in context: A131121 A232853 A328520 * A025527 A334958 A205957

Adjacent sequences: A055769 A055770 A055771 * A055773 A055774 A055775

Keyword

nonn

Author

Labos Elemer, Jul 12 2000

Status

approved