0,3

From Peter Bala, Mar 28 2022: (Start)

The congruence a(n+k) == a(n) (mod k) holds for all n and k.

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.

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)

Seiichi Manyama, Table of n, a(n) for n = 0..430

Peter Bala, Integer sequences that become periodic on reduction modulo k for all k

a(0) = 1 and a(n) = (n-1)! * Sum_{k=1..n} k*A000009(k)*a(n-k)/(n-k)! for n > 0.

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 *)

Cf. A293839, A293841.

Cf. A058892.

Sequence in context: A163605 A294252 A294253 * A327674 A318811 A352299

Adjacent sequences: A293837 A293838 A293839 * A293841 A293842 A293843

nonn,easy

Seiichi Manyama, Oct 17 2017

approved