A062763 -id:A062763 - cp's OEIS frontend

A095996 a(n) = largest divisor of n! that is coprime to n.

1, 1, 2, 3, 24, 5, 720, 315, 4480, 567, 3628800, 1925, 479001600, 868725, 14350336, 638512875, 20922789888000, 14889875, 6402373705728000, 14849255421, 7567605760000, 17717861581875, 1124000727777607680000, 2505147019375

Offset: 1

Robert G. Wilson v, Jul 19 2004, based on a suggestion from Leroy Quet, Jun 18 2004

nonn

The denominators of the coefficients in Taylor series for LambertW(x) are 1, 1, 1, 2, 3, 24, 5, 720, 315, 4480, 567, 3628800, 1925, ..., which is this sequence prefixed by 1. (Cf. A227831.) - N. J. A. Sloane, Aug 02 2013

The second Mathematica program is faster than the first for large n. - T. D. Noe, Sep 07 2013

R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics, Addison-Wesley, 2nd ed., Eq. (5.66).

Vincenzo Librandi, Table of n, a(n) for n = 1..200

Cf. A036503, A227831, A066570.

[Denominator(n^n/Factorial(n)): n in [1..25]]; // Vincenzo Librandi, Sep 04 2014

series(LambertW(x),x,30); # N. J. A. Sloane, Jan 08 2021

f[n_] := Select[Divisors[n! ], GCD[ #, n] == 1 &][[ -1]]; Table[f[n], {n, 30}]
Denominator[Exp[Table[Limit[Zeta[s]*Sum[(1 - If[Mod[k, n] == 0, n, 0])/k^(s - 1), {k, 1, n}], s -> 1], {n, 1, 30}]]] (* Conjecture Mats Granvik, Sep 09 2013 *)
Table[Denominator[n^n/n!], {n, 30}] (* Vincenzo Librandi, Sep 04 2014 *)

a(n):=sum((-1)^(n-j)*binomial(n,j)*(j/n+1)^n,j,0,n);
makelist(num(a(n)),n,1,20); /* Vladimir Kruchinin, Jun 02 2013 */

a(n) = denominator(n^n/n!); \\ G. C. Greubel, Nov 14 2017

a(p) = (p-1)!.
a(n) = n!/A051696(n) = (n-1)!/A062763(n).
a(n) = numerator(Sum_{j = 0..n} (-1)^(n-j)*binomial(n,j)*(j/n+1)^n ). - Vladimir Kruchinin, Jun 02 2013
a(n) = denominator(n^n/n!). - Vincenzo Librandi Sep 04 2014

A051696 Greatest common divisor of n! and n^n.

Original entry on oeis.org

1, 2, 3, 8, 5, 144, 7, 128, 81, 6400, 11, 248832, 13, 100352, 91125, 32768, 17, 429981696, 19, 163840000, 6751269, 63438848, 23, 247669456896, 15625, 1417674752, 1594323, 80564191232, 29, 25076532510720000000, 31, 2147483648

Offset: 1

Leroy Quet

a(n) also equals the smallest positive integer such that lcm(a(1), a(2), a(3), ... a(n)) = n!, for every positive integer n. - Leroy Quet, Apr 28 2007

			a(4) = 8 since 4! = 24 and 4^4 = 256 and gcd(24, 256) = 8.
lcm(a(1), a(2), a(3), a(4), a(5), a(6)) = lcm(1, 2, 3, 8, 5, 144) = 6! = 720. (See comment.)

T. D. Noe, Table of n, a(n) for n = 1..500

seq(igcd(n!, n^n), n=1..32); # Peter Luschny, Nov 29 2017

Table[GCD[n!, n^n], {n, 40}] (* Harvey P. Dale, Oct 20 2011 *)
Table[Numerator@Beta[n, 1/n]/n^(n - 1), {n, 32}] (* Arkadiusz Wesolowski, Nov 22 2017 *)

a(n) = Product_{p|n} p^(sum{k >= 1} floor(n/p^k)), where the product runs over the distinct primes p that divide n. - Leroy Quet, Apr 28 2007
a(n) = n*A062763(n). - R. J. Mathar, Mar 11 2017
a(n) = (numerator of B(n, 1/n))/n^(n - 1), where B(.,.) is the Euler beta function. - Arkadiusz Wesolowski, Nov 22 2017
a(p) = p for p prime. - Peter Luschny, Nov 29 2017

More terms from James Sellers, Dec 08 1999

cp's OEIS Frontend

A095996 a(n) = largest divisor of n! that is coprime to n.

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