A121985 - OEIS

%I #16 Sep 17 2017 22:32:32

%S 12,1,8,81,512,15625,11664,5764801,8388608,387420489,78125000,

%T 3138428376721,5159780352,3937376385699289,21703138331168,

%U 1081219482421875,144115188075855872,14063084452067724991009

%N Denominator of PolyLog(-n, 1/n).

%C 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).

%C 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).

%C PolyLog(-n, 1/n) = A121376(n) / A121985(n) = (Sum_{k=0..n} Eulerian(n,k) * n^(n-k+1)) / (n-1)^(n+1) = n*A122778(n) = (Sum_{k=0..n} Eulerian(n,k) * n^k) / (n-1)^(n+1) = A122020(n) for n > 1.

%H G. C. Greubel, <a href="/A121985/b121985.txt">Table of n, a(n) for n = 1..430</a>

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Polylogarithm.html">Polylogarithm</a>.

%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/EulerianNumber.html">Eulerian number</a>

%F a(n) = denominator(PolyLog(-n, 1/n)).

%F a(n) = denominator((-1)^(n+1) * PolyLog(-n, n)).

%e PolyLog(-n, 1/n) begins -1/12, 6, 33/8, 380/81, 3535/512, 189714/15625, ...

%e a(3) = 2^3;

%e a(4) = 3^4;

%e a(200) = 199^200;

%e a(257) = 2^1809;

%e a(290) = 17^564;

%e a(319) = 2^7 * 3^164 * 53^314, where 2*3*53 = 318 = 319 - 1 and 314 = 319 - 319/53 + 1;

%e a(709) = 2^716 * 3^360 * 59^698;

%e a(710) = 709^710.

%t Join[{12}, Table[Denominator[PolyLog[ -n,1/n]],{n,2,30}]]

%o (PARI) a(n)=denominator(polylog(-n,1/n)) \\ _Charles R Greathouse IV_, Jul 14 2014

%Y Cf. A121376 = numerators.

%Y Cf. A120020, A122778.

%K frac,nonn

%O 1,1

%A _Alexander Adamchuk_, Sep 10 2006, Sep 14 2006