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A013965
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a(n) = sigma_17(n), the sum of the 17th powers of the divisors of n.
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10
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1, 131073, 129140164, 17180000257, 762939453126, 16926788715972, 232630513987208, 2251816993685505, 16677181828806733, 100000762939584198, 505447028499293772, 2218628050709022148, 8650415919381337934, 30491579359845314184, 98526126098761952664
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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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Empirical: Sum_{n>=1} a(n)/exp(2*Pi*n) = 43867/28728, also equals Bernoulli(18)/36. - Simon Plouffe, May 06 2023
Multiplicative with a(p^e) = (p^(17*e+17)-1)/(p^17-1).
Sum_{k=1..n} a(k) = zeta(18) * n^18 / 18 + O(n^19). (End)
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MATHEMATICA
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PROG
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(PARI) my(N=99, q='q+O('q^N)); Vec(sum(n=1, N, n^17*q^n/(1-q^n))) \\ Altug Alkan, Sep 10 2016
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CROSSREFS
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KEYWORD
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nonn,easy,mult
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AUTHOR
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STATUS
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approved
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