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

Author

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

Keywords

• nonn

Comments

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

References

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

Links

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

Crossrefs

Cf. A036503, A227831, A066570.

Programs

Magma

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

Maple

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

Mathematica

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 *)

Maxima

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 */

PARI

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

Formula

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