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A017667
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Numerator of sum of -2nd powers of divisors of n.
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6
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1, 5, 10, 21, 26, 25, 50, 85, 91, 13, 122, 35, 170, 125, 52, 341, 290, 455, 362, 273, 500, 305, 530, 425, 651, 425, 820, 75, 842, 13, 962, 1365, 1220, 725, 52, 637, 1370, 905, 1700, 221, 1682, 625, 1850, 1281, 2366, 1325, 2210, 1705, 2451, 651, 2900, 1785
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OFFSET
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1,2
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COMMENTS
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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
Numerators of coefficients in expansion of Sum_{k>=1} x^k/(k^2*(1 - x^k)). - _Ilya Gutkovskiy_, May 24 2018
C. Defant proves that there are no positive integers n such that sigma_{-2}(n) lies in (Pi^2/8, 5/4). See arxiv link. - _Michel Marcus_, Aug 24 2018
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LINKS
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FORMULA
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Dirichlet g.f.: zeta(s)*zeta(s+2) [for fraction A017667/A017668]. - _Franklin T. Adams-Watters_, Sep 11 2005
sup_{n>=1} a(n)/A017668(n) = zeta(2) (A013661). - _Amiram Eldar_, Sep 25 2022
Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k)/A017668(k) = zeta(3) (A002117). - _Amiram Eldar_, Apr 02 2024
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EXAMPLE
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1, 5/4, 10/9, 21/16, 26/25, 25/18, 50/49, 85/64, 91/81, 13/10, 122/121, 35/24, 170/169, ...
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MATHEMATICA
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Table[Numerator[DivisorSigma[-2, n]], {n, 50}] (* _Vladimir Joseph Stephan Orlovsky_, Jul 21 2011 *)
Table[Numerator[DivisorSigma[2, n]/n^2], {n, 1, 50}] (* _G. C. Greubel_, Nov 08 2018 *)
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PROG
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(PARI) a(n) = numerator(sigma(n, -2)); \\ _Michel Marcus_, Aug 24 2018
(PARI) vector(50, n, numerator(sigma(n, 2)/n^2)) \\ _G. C. Greubel_, Nov 08 2018
(Magma) [Numerator(DivisorSigma(2, n)/n^2): n in [1..50]]; // _G. C. Greubel_, Nov 08 2018
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CROSSREFS
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KEYWORD
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nonn,frac
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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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