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Expansion of Sum_{k>=0} 7^k * x^(7^k) / (1 - x^(7^k))^2.
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%I #17 Jul 14 2023 15:36:30

%S 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 26,27,56,29,30,31,32,33,34,70,36,37,38,39,40,41,84,43,44,45,46,47,48,

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

%F a(n) = n * (A214411(n) + 1).

%F If n == 0 (mod 7), a(n) = n + 7 * a(n/7) otherwise a(n) = n.

%F From _Amiram Eldar_, Jul 14 2023: (Start)

%F Multiplicative with a(7^e) = (e+1)*7^e and a(p^e) = p*e if p != 7.

%F Dirichlet g.f.: (7^s/(7^s-7)) * zeta(s-1).

%F Sum_{k=1..n} a(k) ~ (7/12)*n^2. (End)

%t a[n_] := n * (IntegerExponent[n, 7] + 1); Array[a, 100] (* _Amiram Eldar_, Jul 14 2023 *)

%o (PARI) a(n) = n*(valuation(n, 7)+1);

%Y Cf. A214411, A359099.

%Y Cf. A091512, A364222, A364223.

%K nonn,easy,mult

%O 1,2

%A _Seiichi Manyama_, Jul 14 2023