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A013960
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a(n) = sigma_12(n), the sum of the 12th powers of the divisors of n.
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10
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1, 4097, 531442, 16781313, 244140626, 2177317874, 13841287202, 68736258049, 282430067923, 1000244144722, 3138428376722, 8918294543346, 23298085122482, 56707753666594, 129746582562692, 281543712968705, 582622237229762, 1157115988280531
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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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Vincenzo Librandi, Table of n, a(n) for n = 1..1000
Index entries for sequences related to sigma(n)
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FORMULA
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G.f.: Sum_{k>=1} k^12*x^k/(1-x^k). - Benoit Cloitre, Apr 21 2003
Dirichlet g.f.: zeta(s-12)*zeta(s). - Ilya Gutkovskiy, Sep 10 2016
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MATHEMATICA
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DivisorSigma[12, Range[20]] (* Harvey P. Dale, Jan 28 2015 *)
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PROG
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(Sage) [sigma(n, 12) for n in range(1, 17)] # Zerinvary Lajos, Jun 04 2009
(MAGMA) [DivisorSigma(12, n): n in [1..20]]; // Vincenzo Librandi, Sep 10 2016
(PARI) N=99; q='q+O('q^N); Vec(sum(n=1, N, n^12*q^n/(1-q^n))) \\ Altug Alkan, Sep 10 2016
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CROSSREFS
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Sequence in context: A217196 A321809 A017687 * A036090 A123094 A226695
Adjacent sequences: A013957 A013958 A013959 * A013961 A013962 A013963
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KEYWORD
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nonn,mult,easy
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AUTHOR
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N. J. A. Sloane
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STATUS
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approved
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