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A303281 Expansion of (x/(1 - x)) * (d/dx) Sum_{p prime, k>=1} x^(p^k)/(1 - x^(p^k)).

%I A303281 #23 Jun 13 2025 07:45:51
%S A303281 0,2,5,13,18,30,37,61,79,99,110,146,159,187,217,281,298,352,371,431,
%T A303281 473,517,540,636,686,738,819,903,932,1022,1053,1213,1279,1347,1417,
%U A303281 1561,1598,1674,1752,1912,1953,2079,2122,2254,2389,2481,2528,2768,2866,3016,3118,3274,3327,3543,3653
%N A303281 Expansion of (x/(1 - x)) * (d/dx) Sum_{p prime, k>=1} x^(p^k)/(1 - x^(p^k)).
%C A303281 Sum of exponents in prime-power factorization of hyperfactorial: Product_{k=1..n} k^k (A002109).
%C A303281 Partial sums of A066959.
%H A303281 Amiram Eldar, <a href="/A303281/b303281.txt">Table of n, a(n) for n = 1..10000</a>
%H A303281 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Hyperfactorial.html">Hyperfactorial</a>.
%H A303281 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/K-Function.html">K-Function</a>.
%H A303281 <a href="/index/Eu#epf">Index entries for sequences computed from exponents in factorization of n</a>.
%H A303281 <a href="/index/Fa#factorial">Index entries for sequences related to factorial numbers</a>.
%e A303281 a(4) = 13 because 2^2*3^3*4^4 = 2^10*3^3 and 10 + 3 = 13.
%t A303281 nmax = 55; Rest[CoefficientList[Series[x/(1 - x) D[Sum[Boole[PrimePowerQ[k]] x^k/(1 - x^k), {k, 1, nmax}], x], {x, 0, nmax}], x]]
%t A303281 Table[PrimeOmega[Hyperfactorial[n]], {n, 55}]
%t A303281 Table[Sum[k PrimeOmega[k], {k, n}], {n, 55}]
%t A303281 Accumulate[Table[k * PrimeOmega[k], {k, 1, 55}]] (* _Amiram Eldar_, Jun 13 2025 *)
%o A303281 (PARI) a(n) = sum(k=1, n, k*bigomega(k)); \\ _Altug Alkan_, Apr 20 2018
%Y A303281 Cf. A001222, A002109, A022559, A066959, A303279.
%K A303281 nonn,easy
%O A303281 1,2
%A A303281 _Ilya Gutkovskiy_, Apr 20 2018