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 A294361 #37 Mar 29 2022 02:55:59
%S A294361 1,1,7,43,409,3841,50431,648187,10347793,170363809,3200390551,
%T A294361 62855417131,1371594161257,31147757782753,768384638386639,
%U A294361 19814802390611131,545309251861956001,15661899520801953217,475833949719419469223,15042718034104688144299
%N A294361 E.g.f.: exp(Sum_{n>=1} sigma(n) * x^n).
%C A294361 From _Peter Bala_, Nov 14 2017: (Start)
%C A294361 The terms of the sequence appear to be of the form 6*m + 1.
%C A294361 It appears that the sequence taken modulo 10 is periodic with period 5. 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 A294361 From _Peter Bala_, Mar 28 2022: (Start)
%C A294361 The above conjectures are true. See the Bala link.
%C A294361 a(7*n+2) == 0 (mod 7); a(11*n+9) == 0 (mod 11); a(13*n+11) == 0 (mod 13). (End)
%H A294361 Seiichi Manyama, <a href="/A294361/b294361.txt">Table of n, a(n) for n = 0..430</a>
%H A294361 Peter Bala, <a href="/A047974/a047974_1.pdf">Integer sequences that become periodic on reduction modulo k for all k</a>
%F A294361 a(0) = 1 and a(n) = (n-1)! * Sum_{k=1..n} k*A000203(k)*a(n-k)/(n-k)! for n > 0.
%F A294361 E.g.f.: Product_{k>=1} exp(k*x^k/(1 - x^k)). - _Ilya Gutkovskiy_, Nov 27 2017
%F A294361 a(n) ~ Pi^(1/3) * exp((3*Pi)^(2/3) * n^(2/3)/2 - 3^(1/3) * n^(1/3) / (2*Pi^(2/3)) + 1/24 - 1/(8*Pi^2) - n) * n^(n - 1/6) / 3^(2/3). - _Vaclav Kotesovec_, Sep 04 2018
%t A294361 nmax = 20; CoefficientList[Series[Exp[Sum[DivisorSigma[1, k]*x^k, {k, 1, nmax}]], {x, 0, nmax}], x] * Range[0, nmax]! (* _Vaclav Kotesovec_, Sep 04 2018 *)
%o A294361 (PARI) N=66; x='x+O('x^N); Vec(serlaplace(exp(sum(k=1, N, sigma(k)*x^k))))
%Y A294361 E.g.f.: exp(Sum_{n>=1} sigma_k(n) * x^n): A294363 (k=0), this sequence (k=1), A294362 (k=2).
%Y A294361 Cf. A000203, A053529, A274804, A061256, A192065.
%K A294361 nonn,easy
%O A294361 0,3
%A A294361 _Seiichi Manyama_, Oct 29 2017