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

%I A095996 #48 Apr 30 2025 09:15:03
%S A095996 1,1,2,3,24,5,720,315,4480,567,3628800,1925,479001600,868725,14350336,
%T A095996 638512875,20922789888000,14889875,6402373705728000,14849255421,
%U A095996 7567605760000,17717861581875,1124000727777607680000,2505147019375
%N A095996 a(n) = largest divisor of n! that is coprime to n.
%C A095996 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 A095996 The second Mathematica program is faster than the first for large n. - _T. D. Noe_, Sep 07 2013
%D A095996 R. L. Graham, D. E. Knuth and O. Patashnik, Concrete Mathematics, Addison-Wesley, 2nd ed., Eq. (5.66).
%H A095996 Vincenzo Librandi, <a href="/A095996/b095996.txt">Table of n, a(n) for n = 1..200</a>
%F A095996 a(p) = (p-1)!.
%F A095996 a(n) = n!/A051696(n) = (n-1)!/A062763(n).
%F A095996 a(n) = numerator(Sum_{j = 0..n} (-1)^(n-j)*binomial(n,j)*(j/n+1)^n ). - _Vladimir Kruchinin_, Jun 02 2013
%F A095996 a(n) = denominator(n^n/n!). - _Vincenzo Librandi_ Sep 04 2014
%p A095996 series(LambertW(x),x,30); # _N. J. A. Sloane_, Jan 08 2021
%t A095996 f[n_] := Select[Divisors[n! ], GCD[ #, n] == 1 &][[ -1]]; Table[f[n], {n, 30}]
%t A095996 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 A095996 Table[Denominator[n^n/n!], {n, 30}] (* _Vincenzo Librandi_, Sep 04 2014 *)
%o A095996 (Maxima)
%o A095996 a(n):=sum((-1)^(n-j)*binomial(n,j)*(j/n+1)^n,j,0,n);
%o A095996 makelist(num(a(n)),n,1,20); /* _Vladimir Kruchinin_, Jun 02 2013 */
%o A095996 (Magma) [Denominator(n^n/Factorial(n)): n in [1..25]]; // _Vincenzo Librandi_, Sep 04 2014
%o A095996 (PARI) a(n) = denominator(n^n/n!); \\ _G. C. Greubel_, Nov 14 2017
%Y A095996 Cf. A036503, A227831, A066570.
%K A095996 nonn
%O A095996 1,3
%A A095996 _Robert G. Wilson v_, Jul 19 2004, based on a suggestion from _Leroy Quet_, Jun 18 2004