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A329385 Dirichlet g.f.: 1 / (2 - Product_{k>=1} zeta(k*s)).

%I A329385 #52 Jun 11 2020 16:20:54
%S A329385 1,1,1,3,1,3,1,8,3,3,1,11,1,3,3,22,1,11,1,11,3,3,1,36,3,3,8,11,1,13,1,
%T A329385 59,3,3,3,45,1,3,3,36,1,13,1,11,11,3,1,116,3,11,3,11,1,36,3,36,3,3,1,
%U A329385 57,1,3,11,160,3,13,1,11,3,13,1,164,1,3,11,11,3,13,1,116
%N A329385 Dirichlet g.f.: 1 / (2 - Product_{k>=1} zeta(k*s)).
%H A329385 Vaclav Kotesovec, <a href="/A329385/b329385.txt">Table of n, a(n) for n = 1..100000</a>
%H A329385 Vaclav Kotesovec, <a href="/A329385/a329385.jpg">Graph - The asymptotic ratio (100000 terms)</a>
%F A329385 G.f. A(x) satisfies: A(x) = x + Sum_{k>=2} A000688(k) * A(x^k).
%F A329385 a(1) = 1; a(n) = Sum_{d|n, d < n} A000688(n/d) * a(d).
%F A329385 Let f(s) = Product_{k>=1} zeta(k*s), then Sum_{k=1..n} a(k) ~ n^r / (-r*f'(r)), where r = A335494 = 1.8868691498777... is the root of the equation f(r) = 2 and f'(r) = -1.8255483309672084429580571100367977185868132697213762608374345719289... - _Vaclav Kotesovec_, Jun 11 2020
%t A329385 a[n_] := If[n == 1, n, Sum[If[d < n, FiniteAbelianGroupCount[n/d] a[d], 0], {d, Divisors[n]}]]; Table[a[n], {n, 1, 80}]
%Y A329385 Cf. A000688, A001358 (positions of 3's), A008578 (positions of 1's), A050354, A129667.
%K A329385 nonn
%O A329385 1,4
%A A329385 _Ilya Gutkovskiy_, Jun 07 2020