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

1,2

Comments

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

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.

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

Example

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

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 *)

Prog

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

Crossrefs

Cf. A000142, A007913, A055071, A249831.

Sequence in context: A085087 A072983 A379580 * A212397 A008339 A077139

Adjacent sequences: A055201 A055202 A055203 * A055205 A055206 A055207

Keyword

nonn

Author

Labos Elemer, Jun 19 2000

Status

approved