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A364222 Expansion of Sum_{k>=0} 3^k * x^(3^k) / (1 - x^(3^k))^2.

%I A364222 #17 Jul 14 2023 12:12:41
%S A364222 1,2,6,4,5,12,7,8,27,10,11,24,13,14,30,16,17,54,19,20,42,22,23,48,25,
%T A364222 26,108,28,29,60,31,32,66,34,35,108,37,38,78,40,41,84,43,44,135,46,47,
%U A364222 96,49,50,102,52,53,216,55,56,114,58,59,120,61,62,189,64,65,132,67,68,138,70,71,216,73,74
%N A364222 Expansion of Sum_{k>=0} 3^k * x^(3^k) / (1 - x^(3^k))^2.
%F A364222 a(n) = n * A051064(n).
%F A364222 If n == 0 (mod 3), a(n) = n + 3 * a(n/3) otherwise a(n) = n.
%F A364222 From _Amiram Eldar_, Jul 14 2023: (Start)
%F A364222 Multiplicative with a(3^e) = (e+1)*3^e and a(p^e) = p*e if p != 3.
%F A364222 Dirichlet g.f.: (3^s/(3^s-3)) * zeta(s-1).
%F A364222 Sum_{k=1..n} a(k) ~ (3/4)*n^2. (End)
%t A364222 a[n_] := n * (IntegerExponent[n, 3] + 1); Array[a, 100] (* _Amiram Eldar_, Jul 14 2023 *)
%o A364222 (PARI) a(n) = n*(valuation(n, 3)+1);
%Y A364222 Cf. A007949, A051064, A327625.
%Y A364222 Cf. A091512, A364223, A364224.
%K A364222 nonn,mult,easy
%O A364222 1,2
%A A364222 _Seiichi Manyama_, Jul 14 2023