A294361 E.g.f.: exp(Sum_{n>=1} sigma(n) * x^n).

1, 1, 7, 43, 409, 3841, 50431, 648187, 10347793, 170363809, 3200390551, 62855417131, 1371594161257, 31147757782753, 768384638386639, 19814802390611131, 545309251861956001, 15661899520801953217, 475833949719419469223, 15042718034104688144299

Offset: 0

Author

Seiichi Manyama, Oct 29 2017

Keywords

• nonn
• easy

Comments

From Peter Bala, Nov 14 2017: (Start)

The terms of the sequence appear to be of the form 6*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(7*n+2) == 0 (mod 7); a(11*n+9) == 0 (mod 11); a(13*n+11) == 0 (mod 13). (End)

Links

• 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

Crossrefs

E.g.f.: exp(Sum_{n>=1} sigma_k(n) * x^n): A294363 (k=0), this sequence (k=1), A294362 (k=2).

Cf. A000203, A053529, A274804, A061256, A192065.

Programs

Mathematica

nmax = 20; CoefficientList[Series[Exp[Sum[DivisorSigma[1, k]*x^k, {k, 1, nmax}]], {x, 0, nmax}], x] * Range[0, nmax]! (* Vaclav Kotesovec, Sep 04 2018 *)

PARI

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

Formula

a(0) = 1 and a(n) = (n-1)! * Sum_{k=1..n} k*A000203(k)*a(n-k)/(n-k)! for n > 0.

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

a(n) ~ Pi^(1/3) * exp((3*Pi)^(2/3) * n^(2/3)/2 - 3^(1/3) * n^(1/3) / (2*Pi^(2/3)) + 1/24 - 1/(8*Pi^2) - n) * n^(n - 1/6) / 3^(2/3). - Vaclav Kotesovec, Sep 04 2018