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Denominator of sum of -3rd powers of divisors of n.
3

%I #19 Sep 08 2022 08:44:43

%S 1,8,27,64,125,6,343,512,729,500,1331,432,2197,343,375,4096,4913,648,

%T 6859,4000,1323,2662,12167,384,15625,8788,19683,2744,24389,125,29791,

%U 32768,3993,19652,6125,46656,50653,13718,59319,6400,68921,147,79507,21296,10125

%N Denominator of sum of -3rd powers of divisors of n.

%C 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

%H G. C. Greubel, <a href="/A017670/b017670.txt">Table of n, a(n) for n = 1..10000</a>

%F Denominator of Sum_{d|n} 1/d^3.

%F Denominators of coefficients in expansion of Sum_{k>=1} x^k/(k^3*(1 - x^k)). - _Ilya Gutkovskiy_, May 24 2018

%e 1, 9/8, 28/27, 73/64, 126/125, 7/6, 344/343, 585/512, 757/729, 567/500, 1332/1331, 511/432, ...

%t Table[Denominator[DivisorSigma[-3, n]], {n, 50}] (* _Vladimir Joseph Stephan Orlovsky_, Jul 21 2011 *)

%t Table[Denominator[DivisorSigma[3, n]/n^3], {n, 1, 40}] (* _G. C. Greubel_, Nov 08 2018 *)

%o (PARI) vector(40, n, denominator(sigma(n, 3)/n^3)) \\ _G. C. Greubel_, Nov 08 2018

%o (Magma) [Denominator(DivisorSigma(3,n)/n^3): n in [1..40]]; // _G. C. Greubel_, Nov 08 2018

%Y Cf. A017669.

%K nonn,frac

%O 1,2

%A _N. J. A. Sloane_