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 A013961 a(n) = sigma_13(n), the sum of the 13th powers of the divisors of n. 13
 1, 8193, 1594324, 67117057, 1220703126, 13062296532, 96889010408, 549822930945, 2541867422653, 10001220711318, 34522712143932, 107006334784468, 302875106592254, 793811662272744, 1946196290656824, 4504149450301441, 9904578032905938, 20825519793796029 (list; graph; refs; listen; history; text; internal format)
 OFFSET 1,2 COMMENTS 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 Empirical: Sum_{n>=1} a(n)/exp(2*Pi*n) = 1/24. - Simon Plouffe, Mar 01 2021 LINKS Vincenzo Librandi, Table of n, a(n) for n = 1..1000 FORMULA G.f.: Sum_{k>=1} k^13*x^k/(1-x^k). - Benoit Cloitre, Apr 21 2003 Dirichlet g.f.: zeta(s-13)*zeta(s). - Ilya Gutkovskiy, Sep 10 2016 MAPLE A013961 := proc(n)     numtheory[sigma][13](n) ; end proc: # R. J. Mathar, Sep 21 2017 MATHEMATICA DivisorSigma[13, Range[30]] (* Vincenzo Librandi, Sep 10 2016 *) PROG (Sage) [sigma(n, 13)for n in range(1, 16)] # Zerinvary Lajos, Jun 04 2009 (MAGMA) [DivisorSigma(13, n): n in [1..20]]; // Vincenzo Librandi, Sep 10 2016 (PARI) N=99; q='q+O('q^N); Vec(sum(n=1, N, n^13*q^n/(1-q^n))) \\ Altug Alkan, Sep 10 2016 (PARI) a(n) = sigma(n, 13); \\ Michel Marcus, Sep 10 2016 CROSSREFS Sequence in context: A323544 A230190 A017689 * A036091 A181134 A253713 Adjacent sequences:  A013958 A013959 A013960 * A013962 A013963 A013964 KEYWORD nonn,mult,easy AUTHOR STATUS approved

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Last modified April 19 22:40 EDT 2021. Contains 343117 sequences. (Running on oeis4.)