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A017677
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Numerator of sum of -7th powers of divisors of n.
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3
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1, 129, 2188, 16513, 78126, 23521, 823544, 2113665, 4785157, 5039127, 19487172, 9032611, 62748518, 13279647, 56979896, 270549121, 410338674, 205761751, 893871740, 645047319, 1801914272, 628461297, 3404825448, 385391585, 6103593751, 4047279411, 10465138360, 34691791
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OFFSET
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1,2
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COMMENTS
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Sum_{d|n} 1/d^k is equal to sigma_k(n)/n^k. So sequences A017665-A017712 also give the numerators and denominators of sigma_k(n)/n^k for k = 1..24. The power sums sigma_k(n) are in sequences A000203 (k=1), A001157-A001160 (k=2,3,4,5), A013954-A013972 for k = 6,7,...,24. - Ahmed Fares (ahmedfares(AT)my-deja.com), Apr 05 2001
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LINKS
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FORMULA
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Numerators of coefficients in expansion of Sum_{k>=1} x^k/(k^7*(1 - x^k)). - Ilya Gutkovskiy, May 25 2018
Dirichlet g.f. of a(n)/A017678(n): zeta(s)*zeta(s+7).
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k)/A017678(k) = zeta(8) (A013666). (End)
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EXAMPLE
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1, 129/128, 2188/2187, 16513/16384, 78126/78125, 23521/23328, 823544/823543, 2113665/2097152, ...
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MATHEMATICA
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Table[Numerator[Total[Divisors[n]^-7]], {n, 30}] (* Harvey P. Dale, Nov 29 2014 *)
Table[Numerator[DivisorSigma[7, n]/n^7], {n, 1, 20}] (* G. C. Greubel, Nov 07 2018 *)
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PROG
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(PARI) vector(20, n, numerator(sigma(n, 7)/n^7)) \\ G. C. Greubel, Nov 07 2018
(Magma) [Numerator(DivisorSigma(7, n)/n^7): n in [1..20]]; // G. C. Greubel, Nov 07 2018
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CROSSREFS
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KEYWORD
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nonn,frac
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AUTHOR
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STATUS
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approved
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