|
%I #45 Oct 29 2023 08:36:57
%S 1,8388609,94143178828,70368752566273,11920928955078126,
%T 789730317205170252,27368747340080916344,590295880727458217985,
%U 8862938119746644274757,100000011920928963466734,895430243255237372246532,6624738056749922960468044,41753905413413116367045798
%N a(n) = sigma_23(n), the sum of the 23rd 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 Seiichi Manyama, <a href="/A013971/b013971.txt">Table of n, a(n) for n = 1..10000</a> (first 1000 terms from Harvey P. Dale)
%H <a href="/index/Si#SIGMAN">Index entries for sequences related to sigma(n)</a>.
%F G.f.: Sum_{k>=1} k^23*x^k/(1-x^k). - _Benoit Cloitre_, Apr 21 2003
%F From _Amiram Eldar_, Oct 29 2023: (Start)
%F Multiplicative with a(p^e) = (p^(23*e+23)-1)/(p^23-1).
%F Dirichlet g.f.: zeta(s)*zeta(s-23).
%F Sum_{k=1..n} a(k) = zeta(24) * n^24 / 24 + O(n^25). (End)
%t DivisorSigma[23,Range[15]] (* _Harvey P. Dale_, May 02 2016 *)
%o (Sage) [sigma(n,23)for n in range(1,12)] # _Zerinvary Lajos_, Jun 04 2009
%o (PARI) vector(30, n, sigma(n,23)) \\ _G. C. Greubel_, Nov 03 2018
%o (Magma) [DivisorSigma(23,n): n in [1..30]]; // _G. C. Greubel_, Nov 03 2018
%Y Cf. A000203, A001157-A001160, A013954-A013972, A017665-A017712.
%K nonn,easy,mult
%O 1,2
%A _N. J. A. Sloane_
| 