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%I #13 Feb 12 2022 04:04:55
%S 1,3,3,4,3,9,3,7,4,9,3,12,3,9,9,11,3,12,3,12,9,9,3,21,4,9,7,12,3,27,3,
%T 18,9,9,9,16,3,9,9,21,3,27,3,12,12,9,3,33,4,12,9,12,3,21,9,21,9,9,3,
%U 36,3,9,12,29,9,27,3,12,9,27,3,28,3,9,12,12,9,27,3,33,11,9,3,36,9,9,9
%N Dirichlet g.f.: Product_{p prime} (1 + 2*p^(-s)) / (1 - p^(-s) - p^(-2*s)).
%H Amiram Eldar, <a href="/A351348/b351348.txt">Table of n, a(n) for n = 1..10000</a>
%H Vaclav Kotesovec, <a href="/A351348/a351348.jpg">Graph - the asymptotic ratio (10^8 terms)</a>
%F Multiplicative with a(p^e) = Lucas(e+1).
%F a(n) = Sum_{d|n} A074823(d) * A351219(n/d).
%F From _Vaclav Kotesovec_, Feb 12 2022: (Start)
%F Let f(s) = Product_{p prime} (1 + 1/(p^(2*s) - p^s - 1)) * (1 - 3/p^(2*s) + 2/p^(3*s)), then
%F Sum_{k=1..n} a(k) ~ n * (f(1)*log(n)^2/2 + ((3*g-1)*f(1) + f'(1))*log(n) + (1 - 3*g + 3*g^2 - 3*sg1)*f(1) + (3*g-1)*f'(1) + f''(1)/2), where
%F f(1) = Product_{prime p} (p-1)^3 * (p+2) / (p^2 (p^2 - p - 1)) = 0.76679494740111861346654669603448358442373234633770198438779408968851774...,
%F f'(1) = f(1) * Sum_{p prime} (4*p^2 - 9*p - 4) * log(p) / (p^4 - 4*p^2 + p + 2) = -0.2518173642312369311596467494348076414732211832249275289370643712012051...,
%F f''(1) = f'(1)^2/f(1) + f(1) * Sum_{p prime} -p*(8*p^5 - 27*p^4 - 16*p^3 + 32*p^2 + 16*p + 14) * log(p)^2 / (p^4 - 4*p^2 + p + 2)^2 = 4.28643633804365513728313780779157573071314496047204449783182235740130206...,
%F gamma is the Euler-Mascheroni constant A001620 and sg1 is the first Stieltjes constant (see A082633). (End)
%t f[p_, e_] := LucasL[e + 1]; a[1] = 1; a[n_] := Times @@ f @@@ FactorInteger[n]; Table[a[n], {n, 1, 87}]
%o (PARI) for(n=1, 100, print1(direuler(p=2, n, (1 + 2*X)/(1 - X - X^2))[n], ", ")) \\ _Vaclav Kotesovec_, Feb 10 2022
%Y Cf. A000204, A074823, A351219, A351346, A351347.
%K nonn,mult
%O 1,2
%A _Ilya Gutkovskiy_, Feb 08 2022
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