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A013960 a(n) = sigma_12(n), the sum of the 12th powers of the divisors of n. 10
1, 4097, 531442, 16781313, 244140626, 2177317874, 13841287202, 68736258049, 282430067923, 1000244144722, 3138428376722, 8918294543346, 23298085122482, 56707753666594, 129746582562692, 281543712968705, 582622237229762, 1157115988280531, 2213314919066162 (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
LINKS
FORMULA
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
From Amiram Eldar, Oct 29 2023: (Start)
Multiplicative with a(p^e) = (p^(12*e+12)-1)/(p^12-1).
Sum_{k=1..n} a(k) = zeta(13) * n^13 / 13 + O(n^14). (End)
MATHEMATICA
DivisorSigma[12, Range[20]] (* Harvey P. Dale, Jan 28 2015 *)
PROG
(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) my(N=99, q='q+O('q^N)); Vec(sum(n=1, N, n^12*q^n/(1-q^n))) \\ Altug Alkan, Sep 10 2016
(PARI) a(n) = sigma(n, 12); \\ Amiram Eldar, Oct 29 2023
CROSSREFS
Sequence in context: A342686 A321809 A017687 * A036090 A123094 A226695
KEYWORD
nonn,mult,easy
AUTHOR
STATUS
approved

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Last modified March 29 04:23 EDT 2024. Contains 371264 sequences. (Running on oeis4.)