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A121985
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Denominator of PolyLog(-n, 1/n).
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3
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12, 1, 8, 81, 512, 15625, 11664, 5764801, 8388608, 387420489, 78125000, 3138428376721, 5159780352, 3937376385699289, 21703138331168, 1081219482421875, 144115188075855872, 14063084452067724991009
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OFFSET
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1,1
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COMMENTS
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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).
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.
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LINKS
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FORMULA
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a(n) = denominator(PolyLog(-n, 1/n)).
a(n) = denominator((-1)^(n+1) * PolyLog(-n, n)).
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EXAMPLE
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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.
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MATHEMATICA
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Join[{12}, Table[Denominator[PolyLog[ -n, 1/n]], {n, 2, 30}]]
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PROG
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CROSSREFS
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KEYWORD
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frac,nonn
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AUTHOR
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STATUS
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approved
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