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 A281959 #52 Jan 15 2025 02:55:08
%S A281959 1,33554433,847288609444,1125899940397057,298023223876953126,
%T A281959 28430288877251865252,1341068619663964900808,37778932988857102106625,
%U A281959 717897987692699877379693,10000000298023223910507558,108347059433883722041830252,953962194872104906760006308
%N A281959 a(n) = sigma_25(n), the sum of the 25th powers of the divisors of n.
%C A281959 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). - _Vaclav Kotesovec_, May 07 2023
%C A281959 Since n^25 == n^5 (mod 25), it follows that sigma_25(n) == sigma_5(n) (mod 25). In fact, sigma_25(n) == sigma_(5) (mod 13200), where 13200 = (2^4)*3*(5^2)*11 = A006863(10). - _Peter Bala_, Jan 12 2025
%H A281959 Seiichi Manyama, <a href="/A281959/b281959.txt">Table of n, a(n) for n = 1..10000</a>
%H A281959 Peter Luschny, <a href="https://math.stackexchange.com/questions/4694898/">Is this a new representation of (some) Bernoulli numbers?</a>, Mathematics Stack Exchange.
%H A281959 <a href="/index/Si#SIGMAN">Index entries for sequences related to sigma(n)</a>.
%F A281959 G.f.: Sum_{k>=1} k^25*x^k/(1-x^k).
%F A281959 a(n) == A037947(n) mod 657931.
%F A281959 a(n) = Sum_{k=1..A000005(n)} A275055(k)^25. - _Felix Fröhlich_, Feb 03 2017
%F A281959 Sum_{n>=1} a(n)/exp(2*Pi*n) = 657931/24 = Bernoulli(26)/52. - _Vaclav Kotesovec_, May 07 2023
%F A281959 From _Amiram Eldar_, Oct 29 2023: (Start)
%F A281959 Multiplicative with a(p^e) = (p^(25*e+25)-1)/(p^25-1).
%F A281959 Dirichlet g.f.: zeta(s)*zeta(s-25).
%F A281959 Sum_{k=1..n} a(k) = zeta(26) * n^26 / 26 + O(n^27). (End)
%e A281959 For n = 6: The divisors of 6 are 1, 2, 3, 6, so a(6) = sigma_25(6) = 1^25 + 2^25 + 3^25 + 6^25 = 28430288877251865252. - _Felix Fröhlich_, Feb 03 2017
%t A281959 DivisorSigma[25,Range[20]] (* _Harvey P. Dale_, Jul 08 2024 *)
%o A281959 (PARI) a(n) = sigma(n, 25) \\ _Felix Fröhlich_, Feb 03 2017
%o A281959 (Python)
%o A281959 from sympy import divisor_sigma
%o A281959 def A281959(n): return divisor_sigma(n,25) # _Chai Wah Wu_, May 07 2023
%Y A281959 Cf. A000005, A037947, A275055, A362870.
%K A281959 nonn,mult,easy
%O A281959 1,2
%A A281959 _Seiichi Manyama_, Feb 03 2017