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 A293840 #18 Mar 29 2022 03:40:59
%S A293840 1,1,3,19,121,1041,10651,121843,1575729,22970881,366805171,6365365491,
%T A293840 120044573353,2430782532049,52677233993931,1217023986185491,
%U A293840 29799465317716321,771272544315151233,21044341084622337379,603173026772647474771
%N A293840 E.g.f.: exp(Sum_{n>=1} A000009(n)*x^n).
%C A293840 From _Peter Bala_, Mar 28 2022: (Start)
%C A293840 The congruence a(n+k) == a(n) (mod k) holds for all n and k.
%C A293840 It follows that the sequence obtained by taking a(n) modulo a fixed positive integer k is periodic with exact period dividing k. For example, the sequence taken modulo 10 becomes [1, 1, 3, 9, 1, 1, 1, 3, 9, 1, ...], a purely periodic sequence with exact period 5.
%C A293840 3 divides a(3*n+2); 9 divides a(9*n+8); 11 divides a(11*n+4); 19 divides a(19*n+3). (End)
%H A293840 Seiichi Manyama, <a href="/A293840/b293840.txt">Table of n, a(n) for n = 0..430</a>
%H A293840 Peter Bala, <a href="/A047974/a047974_1.pdf">Integer sequences that become periodic on reduction modulo k for all k</a>
%F A293840 a(0) = 1 and a(n) = (n-1)! * Sum_{k=1..n} k*A000009(k)*a(n-k)/(n-k)! for n > 0.
%t A293840 nmax = 20; CoefficientList[Series[E^Sum[PartitionsQ[k]*x^k, {k, 1, nmax}], {x, 0, nmax}], x] * Range[0, nmax]! (* _Vaclav Kotesovec_, Oct 18 2017 *)
%Y A293840 Cf. A293839, A293841.
%Y A293840 Cf. A058892.
%K A293840 nonn,easy
%O A293840 0,3
%A A293840 _Seiichi Manyama_, Oct 17 2017