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A364224 Expansion of Sum_{k>=0} 7^k * x^(7^k) / (1 - x^(7^k))^2.

%I A364224 #17 Jul 14 2023 15:36:30
%S A364224 1,2,3,4,5,6,14,8,9,10,11,12,13,28,15,16,17,18,19,20,42,22,23,24,25,
%T A364224 26,27,56,29,30,31,32,33,34,70,36,37,38,39,40,41,84,43,44,45,46,47,48,
%U A364224 147,50,51,52,53,54,55,112,57,58,59,60,61,62,126,64,65,66,67,68,69,140,71,72,73,74,75,76,154
%N A364224 Expansion of Sum_{k>=0} 7^k * x^(7^k) / (1 - x^(7^k))^2.
%F A364224 a(n) = n * (A214411(n) + 1).
%F A364224 If n == 0 (mod 7), a(n) = n + 7 * a(n/7) otherwise a(n) = n.
%F A364224 From _Amiram Eldar_, Jul 14 2023: (Start)
%F A364224 Multiplicative with a(7^e) = (e+1)*7^e and a(p^e) = p*e if p != 7.
%F A364224 Dirichlet g.f.: (7^s/(7^s-7)) * zeta(s-1).
%F A364224 Sum_{k=1..n} a(k) ~ (7/12)*n^2. (End)
%t A364224 a[n_] := n * (IntegerExponent[n, 7] + 1); Array[a, 100] (* _Amiram Eldar_, Jul 14 2023 *)
%o A364224 (PARI) a(n) = n*(valuation(n, 7)+1);
%Y A364224 Cf. A214411, A359099.
%Y A364224 Cf. A091512, A364222, A364223.
%K A364224 nonn,easy,mult
%O A364224 1,2
%A A364224 _Seiichi Manyama_, Jul 14 2023