OFFSET
1,1
COMMENTS
PolyLog(n,z) = Sum_{k>=1} z^k/k^n. PolyLog(-n, 1/n) = Sum_{k>=1} k^n/n^k for n > 1. Numerators of PolyLog(-n, 1/n) are listed in A121376(n) = {-1, 6, 33, 380, 3535, 189714, 285929, ...}. a(p+1) = p^(p+1) for prime p. a(p^k+1) = p^( k*p^k + 2*k - (p^k - 1)/(p - 1) ) for prime p and integer k > 0. Prime divisors of a(n) are the same as prime divisors of (n-1).
It appears that for most squarefree (n-1) if q is the largest prime divisor of (n-1) then q^(n - (n-1)/q + 1) divides a(n).
LINKS
G. C. Greubel, Table of n, a(n) for n = 1..430
Eric Weisstein's World of Mathematics, Polylogarithm.
Eric Weisstein's World of Mathematics, Eulerian number
FORMULA
a(n) = denominator(PolyLog(-n, 1/n)).
a(n) = denominator((-1)^(n+1) * PolyLog(-n, n)).
EXAMPLE
PolyLog(-n, 1/n) begins -1/12, 6, 33/8, 380/81, 3535/512, 189714/15625, ...
a(3) = 2^3;
a(4) = 3^4;
a(200) = 199^200;
a(257) = 2^1809;
a(290) = 17^564;
a(319) = 2^7 * 3^164 * 53^314, where 2*3*53 = 318 = 319 - 1 and 314 = 319 - 319/53 + 1;
a(709) = 2^716 * 3^360 * 59^698;
a(710) = 709^710.
MATHEMATICA
Join[{12}, Table[Denominator[PolyLog[ -n, 1/n]], {n, 2, 30}]]
PROG
(PARI) a(n)=denominator(polylog(-n, 1/n)) \\ Charles R Greathouse IV, Jul 14 2014
CROSSREFS
KEYWORD
frac,nonn
AUTHOR
Alexander Adamchuk, Sep 10 2006, Sep 14 2006
STATUS
approved