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A364223 Expansion of Sum_{k>=0} 5^k * x^(5^k) / (1 - x^(5^k))^2.

%I A364223 #15 Jul 15 2023 05:54:06
%S A364223 1,2,3,4,10,6,7,8,9,20,11,12,13,14,30,16,17,18,19,40,21,22,23,24,75,
%T A364223 26,27,28,29,60,31,32,33,34,70,36,37,38,39,80,41,42,43,44,90,46,47,48,
%U A364223 49,150,51,52,53,54,110,56,57,58,59,120,61,62,63,64,130,66,67,68,69,140,71,72,73,74,225
%N A364223 Expansion of Sum_{k>=0} 5^k * x^(5^k) / (1 - x^(5^k))^2.
%F A364223 a(n) = n * A055457(n).
%F A364223 If n == 0 (mod 5), a(n) = n + 5 * a(n/5) otherwise a(n) = n.
%F A364223 From _Amiram Eldar_, Jul 14 2023: (Start)
%F A364223 Multiplicative with a(5^e) = (e+1)*5^e and a(p^e) = p*e if p != 5.
%F A364223 Dirichlet g.f.: (5^s/(5^s-5)) * zeta(s-1).
%F A364223 Sum_{k=1..n} a(k) ~ (5/8)*n^2. (End)
%t A364223 a[n_] := n * (IntegerExponent[n, 5] + 1); Array[a, 100] (* _Amiram Eldar_, Jul 14 2023 *)
%o A364223 (PARI) a(n) = n*(valuation(n, 5)+1);
%Y A364223 Cf. A055457, A112765, A359100.
%Y A364223 Cf. A091512, A364222, A364224.
%K A364223 nonn,easy,mult
%O A364223 1,2
%A A364223 _Seiichi Manyama_, Jul 14 2023