A294361 - OEIS

%I #37 Mar 29 2022 02:55:59

%S 1,1,7,43,409,3841,50431,648187,10347793,170363809,3200390551,

%T 62855417131,1371594161257,31147757782753,768384638386639,

%U 19814802390611131,545309251861956001,15661899520801953217,475833949719419469223,15042718034104688144299

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

%C From _Peter Bala_, Nov 14 2017: (Start)

%C The terms of the sequence appear to be of the form 6*m + 1.

%C 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)

%C From _Peter Bala_, Mar 28 2022: (Start)

%C The above conjectures are true. See the Bala link.

%C a(7*n+2) == 0 (mod 7); a(11*n+9) == 0 (mod 11); a(13*n+11) == 0 (mod 13). (End)

%H Seiichi Manyama, <a href="/A294361/b294361.txt">Table of n, a(n) for n = 0..430</a>

%H Peter Bala, <a href="/A047974/a047974_1.pdf">Integer sequences that become periodic on reduction modulo k for all k</a>

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

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

%F 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

%t 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 *)

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

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

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

%K nonn,easy

%O 0,3

%A _Seiichi Manyama_, Oct 29 2017