%I #49 Oct 29 2023 02:39:02

%S 1,131073,129140164,17180000257,762939453126,16926788715972,

%T 232630513987208,2251816993685505,16677181828806733,

%U 100000762939584198,505447028499293772,2218628050709022148,8650415919381337934,30491579359845314184,98526126098761952664

%N a(n) = sigma_17(n), the sum of the 17th powers of the divisors of n.

%C 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).

%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 Vincenzo Librandi, <a href="/A013965/b013965.txt">Table of n, a(n) for n = 1..1000</a>

%H <a href="/index/Si#SIGMAN">Index entries for sequences related to sigma(n)</a>.

%F G.f.: Sum_{k>=1} k^17*x^k/(1-x^k). - _Benoit Cloitre_, Apr 21 2003

%F Dirichlet g.f.: zeta(s-17)*zeta(s). - _Ilya Gutkovskiy_, Sep 10 2016

%F Empirical: Sum_{n>=1} a(n)/exp(2*Pi*n) = 43867/28728, also equals Bernoulli(18)/36. - _Simon Plouffe_, May 06 2023

%F From _Amiram Eldar_, Oct 29 2023: (Start)

%F Multiplicative with a(p^e) = (p^(17*e+17)-1)/(p^17-1).

%F Sum_{k=1..n} a(k) = zeta(18) * n^18 / 18 + O(n^19). (End)

%t DivisorSigma[17,Range[20]] (* _Harvey P. Dale_, May 30 2013 *)

%o (Sage) [sigma(n,17)for n in range(1,14)] # _Zerinvary Lajos_, Jun 04 2009

%o (Magma) [DivisorSigma(17, n): n in [1..20]]; // _Vincenzo Librandi_, Sep 10 2016

%o (PARI) my(N=99, q='q+O('q^N)); Vec(sum(n=1, N, n^17*q^n/(1-q^n))) \\ _Altug Alkan_, Sep 10 2016

%o (PARI) a(n) = sigma(n, 17); \\ _Amiram Eldar_, Oct 29 2023

%Y Cf. A000203, A001157-A001160, A013676, A013954-A013972, A017665-A017712.

%K nonn,easy,mult

%O 1,2

%A _N. J. A. Sloane_