%I #13 Apr 23 2018 08:52:16
%S 0,2,5,13,18,30,37,61,79,99,110,146,159,187,217,281,298,352,371,431,
%T 473,517,540,636,686,738,819,903,932,1022,1053,1213,1279,1347,1417,
%U 1561,1598,1674,1752,1912,1953,2079,2122,2254,2389,2481,2528,2768,2866,3016,3118,3274,3327,3543,3653
%N Expansion of (x/(1 - x)) * (d/dx) Sum_{p prime, k>=1} x^(p^k)/(1 - x^(p^k)).
%C Sum of exponents in prime-power factorization of hyperfactorial: Product_{k=1..n} k^k (A002109).
%C Partial sums of A066959.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/Hyperfactorial.html">Hyperfactorial</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/K-Function.html">K-Function</a>
%H <a href="/index/Fa#factorial">Index entries for sequences related to factorial numbers</a>
%H <a href="/index/Eu#epf">Index entries for sequences computed from exponents in factorization of n</a>
%e a(4) = 13 because 2^2*3^3*4^4 = 2^10*3^3 and 10 + 3 = 13.
%t 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 Table[PrimeOmega[Hyperfactorial[n]], {n, 55}]
%t Table[Sum[k PrimeOmega[k], {k, n}], {n, 55}]
%o (PARI) a(n) = sum(k=1, n, k*bigomega(k)); \\ _Altug Alkan_, Apr 20 2018
%Y Cf. A001222, A002109, A022559, A066959, A303279.
%K nonn
%O 1,2
%A _Ilya Gutkovskiy_, Apr 20 2018