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 A294362 #36 Mar 29 2022 02:56:24
%S A294362 1,1,11,91,1105,13841,230731,3955771,80483201,1738065025,41800101931,
%T A294362 1070731623611,29804263624081,878224530964561,27672361220570795,
%U A294362 919409968480087771,32304618825218432641,1191168445737728717441,46119903359374012564171
%N A294362 E.g.f.: exp(Sum_{n>=1} sigma_2(n) * x^n).
%C A294362 From _Peter Bala_, Nov 14 2017: (Start)
%C A294362 It appears that the sequence taken modulo 10 is periodic with period (1, 1, 1, 1, 5) of length 5.
%C A294362 More generally, we conjecture that for k = 2,3,4,... the sequence a(n+k) - a(n) is divisible by k: if true, then for each k the sequence a(n) taken modulo k would be periodic with the exact period dividing k. (End)
%C A294362 From _Peter Bala_, Mar 28 2022: (Start)
%C A294362 The above conjectures are true. See the Bala link.
%C A294362 a(5*n+4) = 0 (mod 5); a(7*n+3) == 0 (mod 7); a(11*n+2) == 0 (mod 11); a(13*n+3) == 0 (mod 13); a(17*n+4) == 0 (mod 17); a(19*n+12) == 0 (mod 19). (End)
%H A294362 Seiichi Manyama, <a href="/A294362/b294362.txt">Table of n, a(n) for n = 0..417</a>
%H A294362 Peter Bala, <a href="/A047974/a047974_1.pdf">Integer sequences that become periodic on reduction modulo k for all k</a>
%F A294362 a(0) = 1 and a(n) = (n-1)! * Sum_{k=1..n} k*A001157(k)*a(n-k)/(n-k)! for n > 0.
%F A294362 E.g.f.: Product_{k>=1} exp(k^2*x^k/(1 - x^k)). - _Ilya Gutkovskiy_, Nov 27 2017
%F A294362 a(n) ~ (3*Zeta(3))^(1/8) * exp(2^(9/4) * Zeta(3)^(1/4) * n^(3/4) / 3^(3/4) - n^(1/4) / (2^(9/4) * 3^(5/4) * Zeta(3)^(1/4)) - n) * n^(n - 1/8) / 2^(7/8). - _Vaclav Kotesovec_, Sep 04 2018
%t A294362 nmax = 20; CoefficientList[Series[Exp[Sum[DivisorSigma[2, k]*x^k, {k, 1, nmax}]], {x, 0, nmax}], x] * Range[0, nmax]! (* _Vaclav Kotesovec_, Sep 04 2018 *)
%o A294362 (PARI) N=66; x='x+O('x^N); Vec(serlaplace(exp(sum(k=1, N, sigma(k, 2)*x^k))))
%Y A294362 E.g.f.: exp(Sum_{n>=1} sigma_k(n) * x^n): A294363 (k=0), A294361 (k=1), this sequence (k=2).
%Y A294362 Cf. A001157.
%K A294362 nonn,easy
%O A294362 0,3
%A A294362 _Seiichi Manyama_, Oct 29 2017