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

Author

Seiichi Manyama, Oct 29 2017

Keywords

• nonn
• easy

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

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.

Programs

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 *)

PARI

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