A294363 E.g.f.: exp(Sum_{n>=1} d(n) * x^n), where d(n) is the number of divisors of n.

1, 1, 5, 25, 193, 1481, 16021, 167665, 2220065, 30004273, 468585541, 7560838121, 138355144225, 2589359765305, 53501800316693, 1146089983207681, 26457132132638401, 632544682981967585, 16171678558995779845, 426926324177655018553, 11938570457328874969601

Offset

0,3

Comments

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)

Links

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

Formula

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

Mathematica

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 *)

Prog

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

Crossrefs

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

Keyword

nonn,easy

Author

Seiichi Manyama, Oct 29 2017

Status

approved