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%I #42 Sep 08 2022 08:45:14
%S 1,1,2,3,24,5,720,315,4480,567,3628800,1925,479001600,868725,14350336,
%T 638512875,20922789888000,14889875,6402373705728000,14849255421,
%U 7567605760000,17717861581875,1124000727777607680000,2505147019375
%N a(n) = largest divisor of n! that is coprime to n.
%C 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
%C The second Mathematica program is faster than the first for large n. - _T. D. Noe_, Sep 07 2013
%D R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics, Addison-Wesley, 2nd ed., Eq. (5.66).
%H Vincenzo Librandi, <a href="/A095996/b095996.txt">Table of n, a(n) for n = 1..200</a>
%F a(p) = (p-1)!.
%F a(n) = n!/A051696(n) = (n-1)!/A062763(n).
%F a(n) = numerator(Sum_{j = 0..n} (-1)^(n-j)*binomial(n,j)*(j/n+1)^n ). - _Vladimir Kruchinin_, Jun 02 2013
%F a(n) = denominator(n^n/n!). - _Vincenzo Librandi_ Sep 04 2014
%p series(LambertW(x),x,30); # _N. J. A. Sloane_, Jan 08 2021
%t f[n_] := Select[Divisors[n! ], GCD[ #, n] == 1 &][[ -1]]; Table[f[n], {n, 30}]
%t 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 *)
%t Table[Denominator[n^n/n!], {n, 30}] (* _Vincenzo Librandi_, Sep 04 2014 *)
%o (Maxima)
%o a(n):=sum((-1)^(n-j)*binomial(n,j)*(j/n+1)^n,j,0,n);
%o makelist(num(a(n),n,1,20); /* _Vladimir Kruchinin_, Jun 02 2013 */
%o (Magma) [Denominator(n^n/Factorial(n)): n in [1..25]]; // _Vincenzo Librandi_, Sep 04 2014
%o (PARI) for(n=1,50, print1(denominator(n^n/n!), ", ")) \\ _G. C. Greubel_, Nov 14 2017
%Y Cf. A036503, A227831, A066570.
%K nonn
%O 1,3
%A _Robert G. Wilson v_, Jul 19 2004, based on a suggestion from _Leroy Quet_, Jun 18 2004
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