0,3

From Peter Bala, Nov 13 2017: (Start)

The terms of the sequence appear to be of the form 4*m + 1.

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)

From Peter Bala, Mar 28 2022: (Start)

The above conjectures are true. See the Bala link.

a(5*n+2) == 0 (mod 5); a(5*n+3) == 0 (mod 5); a(13*n+9) == 0 (mod 13). (End)

Seiichi Manyama, Table of n, a(n) for n = 0..438

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*A000005(k)*a(n-k)/(n-k)! for n > 0.

E.g.f.: Product_{k>=1} exp(x^k/(1 - x^k)). - Ilya Gutkovskiy, Nov 27 2017

Conjecture: log(a(n)/n!) ~ sqrt(2*n*log(n)). - Vaclav Kotesovec, Sep 07 2018

nmax = 20; CoefficientList[Series[Exp[Sum[DivisorSigma[0, k]*x^k, {k, 1, nmax}]], {x, 0, nmax}], x] * Range[0, nmax]! (* Vaclav Kotesovec, Sep 05 2018 *)

a[n_] := a[n] = If[n == 0, 1, Sum[k*DivisorSigma[0, k]*a[n-k], {k, 1, n}]/n]; Table[n!*a[n], {n, 0, 20}] (* Vaclav Kotesovec, Sep 06 2018 *)

(PARI) N=66; x='x+O('x^N); Vec(serlaplace(exp(sum(k=1, N, numdiv(k)*x^k))))

E.g.f.: exp(Sum_{n>=1} sigma_k(n) * x^n): this sequence (k=0), A294361 (k=1), A294362 (k=2).

Cf. A000005, A028342, A295739, A006171, A107742.

Sequence in context: A096684 A265184 A096245 * A294213 A330198 A346269

Adjacent sequences: A294360 A294361 A294362 * A294364 A294365 A294366

nonn,easy

Seiichi Manyama, Oct 29 2017

approved