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%I #39 Mar 29 2022 03:42:30
%S 1,1,5,25,193,1481,16021,167665,2220065,30004273,468585541,7560838121,
%T 138355144225,2589359765305,53501800316693,1146089983207681,
%U 26457132132638401,632544682981967585,16171678558995779845,426926324177655018553,11938570457328874969601
%N E.g.f.: exp(Sum_{n>=1} d(n) * x^n), where d(n) is the number of divisors of n.
%C From _Peter Bala_, Nov 13 2017: (Start)
%C The terms of the sequence appear to be of the form 4*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(5*n+2) == 0 (mod 5); a(5*n+3) == 0 (mod 5); a(13*n+9) == 0 (mod 13). (End)
%H Seiichi Manyama, <a href="/A294363/b294363.txt">Table of n, a(n) for n = 0..438</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*A000005(k)*a(n-k)/(n-k)! for n > 0.
%F E.g.f.: Product_{k>=1} exp(x^k/(1 - x^k)). - _Ilya Gutkovskiy_, Nov 27 2017
%F Conjecture: log(a(n)/n!) ~ sqrt(2*n*log(n)). - _Vaclav Kotesovec_, Sep 07 2018
%t 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 *)
%t 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 *)
%o (PARI) N=66; x='x+O('x^N); Vec(serlaplace(exp(sum(k=1, N, numdiv(k)*x^k))))
%Y E.g.f.: exp(Sum_{n>=1} sigma_k(n) * x^n): this sequence (k=0), A294361 (k=1), A294362 (k=2).
%Y Cf. A000005, A028342, A295739, A006171, A107742.
%K nonn,easy
%O 0,3
%A _Seiichi Manyama_, Oct 29 2017
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