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Numerator of sum of -6th powers of divisors of n.
3

%I #31 Apr 02 2024 02:55:05

%S 1,65,730,4161,15626,23725,117650,266305,532171,101569,1771562,506255,

%T 4826810,3823625,2281396,17043521,24137570,34591115,47045882,32509893,

%U 85884500,57575765,148035890,97201325,244156251,12067025,387952660,244770825,594823322

%N Numerator of sum of -6th 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="/A017675/b017675.txt">Table of n, a(n) for n = 1..10000</a>

%F Numerators of coefficients in expansion of Sum_{k>=1} x^k/(k^6*(1 - x^k)). - _Ilya Gutkovskiy_, May 25 2018

%F From _Amiram Eldar_, Apr 02 2024: (Start)

%F sup_{n>=1} a(n)/A017676(n) = zeta(6) (A013664).

%F Dirichlet g.f. of a(n)/A017676(n): zeta(s)*zeta(s+6).

%F Asymptotic mean: Limit_{m->oo} (1/m) * Sum_{k=1..m} a(k)/A017676(k) = zeta(7) (A013665). (End)

%e 1, 65/64, 730/729, 4161/4096, 15626/15625, 23725/23328, 117650/117649, 266305/262144, ...

%t A017675[n_Integer] := Numerator[DivisorSigma[-6, n]]; Table[A017675[n], {n, 100}] (* _Vladimir Joseph Stephan Orlovsky_, Jul 22 2011 *)

%t Table[Numerator[DivisorSigma[6, n]/n^6], {n, 1, 20}] (* _G. C. Greubel_, Nov 07 2018 *)

%o (PARI) vector(20, n, numerator(sigma(n, 6)/n^6)) \\ _G. C. Greubel_, Nov 07 2018

%o (Magma) [Numerator(DivisorSigma(6,n)/n^6): n in [1..20]]; // _G. C. Greubel_, Nov 07 2018

%Y Cf. A017676 (denominator), A013664, A013665.

%K nonn,frac

%O 1,2

%A _N. J. A. Sloane_