This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.
%I A362870 #45 Jan 15 2025 03:00:17
%S A362870 1,536870913,68630377364884,288230376688582657,186264514923095703126,
%T A362870 36845653355419807219092,3219905755813179726837608,
%U A362870 154742505198902911050973185,4710128697246313465298968573,100000000186264514923632574038,1586309297171491574414436704892
%N A362870 a(n) = sigma_29(n), the sum of the 29th powers of the divisors of n.
%C A362870 In general, for k > 0, Sum_{n>=1} sigma_(4*k+1)(n) / exp(2*Pi*n) = Bernoulli(4*k+2)/(8*k+4). For k = 0, Sum_{n>=1} sigma(n)/exp(2*Pi*n) = 1/24 - 1/(8*Pi) = Bernoulli(2)/4 - 1/(8*Pi).
%C A362870 This formula can best be understood as a statement about the divided Bernoulli numbers b(n) = B(n) / n. Then you can say: If v is twice an odd number greater than 1 (i.e., v = 4*n + 2, a term of A016825 that is greater than 2), then b(v) = 2 * Sum_{j>=1} sigma_{v - 1}(j) / exp(2*Pi*j) = A358625(v) / A075180(v - 1). - _Peter Luschny_, May 08 2023
%H A362870 Seiichi Manyama, <a href="/A362870/b362870.txt">Table of n, a(n) for n = 1..10000</a>
%H A362870 Peter Luschny, <a href="https://math.stackexchange.com/questions/4694898/">Is this a new representation of (some) Bernoulli numbers?</a>, Mathematics Stack Exchange.
%H A362870 <a href="/index/Si#SIGMAN">Index entries for sequences related to sigma(n)</a>.
%F A362870 G.f.: Sum_{k>=1} k^29 * x^k / (1-x^k).
%F A362870 Dirichlet g.f.: zeta(s-29)*zeta(s).
%F A362870 Sum_{k=1..n} a(k) ~ zeta(30) * n^30 / 30.
%F A362870 Sum_{n>=1} a(n)/exp(2*Pi*n) = 1723168255201/171864 = Bernoulli(30)/60.
%F A362870 Multiplicative with a(p^e) = (p^(29*e+29)-1)/(p^29-1). - _Amiram Eldar_, Oct 29 2023
%p A362870 with(NumberTheory): seq(SumOfDivisors(k, 29), k = 1..20);
%t A362870 DivisorSigma[29, Range[20]]
%o A362870 (PARI) for(n=1, 20, print1(direuler( p=2, n, 1 / (1 - X) /(1 - p^29*X))[n], ", "))
%o A362870 (Python)
%o A362870 from sympy import divisor_sigma
%o A362870 def A362870(n): return divisor_sigma(n,29) # _Chai Wah Wu_, May 07 2023
%Y A362870 Cf. A000203 (sigma_1), A001160 (sigma_5), A013957 (sigma_9), A013961 (sigma_13), A013965 (sigma_17), A013969 (sigma_21), A281959 (sigma_25).
%Y A362870 Cf. A075180, A358625, A016825, A082771.
%K A362870 nonn,easy,mult
%O A362870 1,2
%A A362870 _Vaclav Kotesovec_, May 07 2023