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a(n) = sigma_21(n), the sum of the 21st powers of the divisors of n.
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%I #38 Oct 29 2023 08:36:41

%S 1,2097153,10460353204,4398048608257,476837158203126,

%T 21936961102828212,558545864083284008,9223376434903384065,

%U 109418989141972712413,1000000476837160300278,7400249944258160101212,46005141850728850805428,247064529073450392704414,1171356134499851307229224

%N a(n) = sigma_21(n), the sum of the 21st 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="/A013969/b013969.txt">Table of n, a(n) for n = 1..10000</a>

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

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

%F Sum_{n>=1} a(n)/exp(2*Pi*n) = 77683/552 = Bernoulli(22)/44. - _Vaclav Kotesovec_, May 07 2023

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

%F Multiplicative with a(p^e) = (p^(21*e+21)-1)/(p^21-1).

%F Dirichlet g.f.: zeta(s)*zeta(s-21).

%F Sum_{k=1..n} a(k) = zeta(22) * n^22 / 22 + O(n^23). (End)

%t Table[DivisorSigma[21,n],{n,50}] (* _Vladimir Joseph Stephan Orlovsky_, Mar 11 2009 *)

%o (Sage) [sigma(n,21)for n in range(1,13)] # _Zerinvary Lajos_, Jun 04 2009

%o (PARI) vector(50, n, sigma(n,21)) \\ _G. C. Greubel_, Nov 03 2018

%o (Magma) [DivisorSigma(21,n): n in [1..50]]; // _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_