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A013956
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a(n) = sigma_8(n), the sum of the 8th powers of the divisors of n.
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14
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1, 257, 6562, 65793, 390626, 1686434, 5764802, 16843009, 43053283, 100390882, 214358882, 431733666, 815730722, 1481554114, 2563287812, 4311810305, 6975757442, 11064693731, 16983563042, 25700456418, 37828630724, 55090232674, 78310985282, 110523825058, 152588281251
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OFFSET
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1,2
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COMMENTS
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If the canonical factorization of n into prime powers is the product of p^e(p) then sigma_k(n) = Product_p ((p^((e(p)+1)*k))-1)/(p^k-1).
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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L.g.f.: -log(Product_{k>=1} (1 - x^k)^(k^7)) = Sum_{n>=1} a(n)*x^n/n. - Ilya Gutkovskiy, May 06 2017
Multiplicative with a(p^e) = (p^(8*e+8)-1)/(p^8-1).
Dirichlet g.f.: zeta(s)*zeta(s-8).
Sum_{k=1..n} a(k) = zeta(9) * n^9 / 9 + O(n^10). (End)
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MATHEMATICA
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PROG
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(Magma) [DivisorSigma(8, n): n in [1..30]]; // Bruno Berselli, Apr 10 2013
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CROSSREFS
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KEYWORD
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nonn,mult,easy
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AUTHOR
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STATUS
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approved
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