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a(n) = sigma_9(n), the sum of the 9th powers of the divisors of n.
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%I #54 Oct 29 2023 02:57:22

%S 1,513,19684,262657,1953126,10097892,40353608,134480385,387440173,

%T 1001953638,2357947692,5170140388,10604499374,20701400904,38445332184,

%U 68853957121,118587876498,198756808749,322687697780,513002215782,794320419872,1209627165996,1801152661464

%N a(n) = sigma_9(n), the sum of the 9th 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

%C Note that the sequence is not monotonically increasing, with a(4488) > a(4489) being the first of infinitely many examples. - _Charles R Greathouse IV_, Dec 28 2021

%H Vincenzo Librandi, <a href="/A013957/b013957.txt">Table of n, a(n) for n = 1..1000</a>

%H T. H. Grönwall, <a href="https://www.jstor.org/stable/1988773">Some asymptotic expressions in the Theory of Numbers</a>, Trans. Amer. Math. Soc., Vol. 14, No. 1 (1913), pp. 113-122.

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

%F G.f.: Sum_{k>=1} k^9*x^k/(1-x^k). - _Benoit Cloitre_, Apr 21 2003

%F L.g.f.: -log(Product_{k>=1} (1 - x^k)^(k^8)) = Sum_{n>=1} a(n)*x^n/n. - _Ilya Gutkovskiy_, May 06 2017

%F n^9 + 1 <= a(n) < zeta(9)*n^9. In particular, Grönwall proves lim sup a(n)/n^9 = zeta(9) = A013667. - _Charles R Greathouse IV_, Dec 27 2021

%F Sum_{n>=1} a(n)/exp(2*Pi*n) = 1/264 = Bernoulli(10)/20. - _Vaclav Kotesovec_, May 07 2023

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

%F Multiplicative with a(p^e) = (p^(9*e+9)-1)/(p^9-1).

%F Dirichlet g.f.: zeta(s)*zeta(s-9).

%F Sum_{k=1..n} a(k) = zeta(10) * n^10 / 10 + O(n^11). (End)

%t Table[DivisorSigma[9,n],{n,50}] (* _Vladimir Joseph Stephan Orlovsky_, Mar 11 2009 *)

%o (PARI) a(n)=if(n<1,0,sigma(n,9))

%o (Sage) [sigma(n,9)for n in range(1,21)] # _Zerinvary Lajos_, Jun 04 2009

%o (Magma) [DivisorSigma(9,n): n in [1..20]]; // _Bruno Berselli_, Apr 10 2013

%Y Cf. A000203, A001157-A001160, A013667, A013668, A013954-A013972, A017665-A017712.

%K nonn,mult,easy

%O 1,2

%A _N. J. A. Sloane_