A249831 -id:A249831 - cp's OEIS frontend

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

Labos Elemer, Jun 19 2000

nonn

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

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

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.

Cf. A000142, A007913, A055071, A249831.

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 *)

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

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

A008339 a(1)=1; for n >= 1, a(n+1) = lcm(a(n),n) / gcd(a(n),n).

Original entry on oeis.org

1, 1, 2, 6, 6, 30, 5, 35, 280, 2520, 252, 2772, 231, 3003, 858, 1430, 5720, 97240, 437580, 8314020, 415701, 969969, 176358, 4056234, 2704156, 67603900, 2600150, 70204050, 10029150, 290845350, 9694845, 300540195, 9617286240, 35263382880, 1037158320

Offset: 1

N. J. A. Sloane

nonn

a(n+1) = Product(A098666(n,k): 1<=k<=n), row-products of triangle A098666. - Reinhard Zumkeller, Sep 20 2004

a(n+1) is divisible by all primes in (n/2, n]; thus lim_{n->infinity} a(n) = infinity. - Franklin T. Adams-Watters, Dec 13 2006

Vincenzo Librandi, Table of n, a(n) for n = 1..1000

Cf. A249831.

A008339 := proc(n) option remember; if n = 1 then 1 else lcm(A008339(n-1), n-1)/gcd(A008339(n-1), n-1); fi; end;

FoldList[ LCM[ #1, #2 ]/GCD[ #1, #2 ]&, 1, Range[ 30 ] ] (* Olivier Gérard, Aug 15 1997 *)

a(1) = 1, a(n) = a(n-1)*r/s where y is the largest divisor of a(n-1) with r*s = n. - Amarnath Murthy, Jul 01 2003

a(1) = 1, a(n+1) = a(n)*n/gcd(a(n),n)^2. - Franklin T. Adams-Watters, Dec 13 2006

cp's OEIS Frontend

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

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