A055204 Squarefree part of n!: n! divided by its largest square divisor.

1, 2, 6, 6, 30, 5, 35, 70, 70, 7, 77, 231, 3003, 858, 1430, 1430, 24310, 12155, 230945, 46189, 969969, 176358, 4056234, 676039, 676039, 104006, 312018, 44574, 1292646, 1077205, 33393355, 66786710, 2203961430, 64822395, 90751353, 90751353

Offset: 1

Author

Labos Elemer, Jun 19 2000

Keywords

• nonn

Comments

Smallest number such that n!*a(n) is a square.

Examples

			10! = 518400*7 = 7*(720)^2, so a(10) = 7.
		

Links

• Charles R Greathouse IV, Table of n, a(n) for n = 1..3386 (next term has 1001 digits)
• Kevin A. Broughan, Asymptotic Order of the Square-free Part of N!, Integers, 2 (2002), Article A.10.
• Rafael Jakimczuk, On the h-th free part of the factorial, International Mathematical Forum, Vol. 12, No. 13 (2017), pp. 629-634.

Crossrefs

Cf. A000142, A007913, A055071, A249831.

Programs

Mathematica

f[p_, e_] := p^Mod[e, 2]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n!]; Array[a, 40] (* Amiram Eldar, Sep 01 2024 *)
a[n_] := Block[{fi = Transpose@ FactorInteger[n!]}, Times @@ (fi[[1]]^Mod[fi[[2]], 2])]; Array[a, 40] (* Robert G. Wilson v, Nov 17 2024 *)

PARI

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

Formula

a(n) = A007913(n!) = n!/A055071(n) = A000142(n)/A055071(n).

log a(n) ~ n log 2. - Charles R Greathouse IV, Apr 03 2012

sqrt(n!) = A055772(n) * sqrt(a(n)). - Alonso del Arte, Feb 16 2015