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

0, 2, 5, 13, 18, 30, 37, 61, 79, 99, 110, 146, 159, 187, 217, 281, 298, 352, 371, 431, 473, 517, 540, 636, 686, 738, 819, 903, 932, 1022, 1053, 1213, 1279, 1347, 1417, 1561, 1598, 1674, 1752, 1912, 1953, 2079, 2122, 2254, 2389, 2481, 2528, 2768, 2866, 3016, 3118, 3274, 3327, 3543, 3653

Offset: 1

Author

Ilya Gutkovskiy, Apr 20 2018

Keywords

• nonn
• easy

Comments

Sum of exponents in prime-power factorization of hyperfactorial: Product_{k=1..n} k^k (A002109).

Partial sums of A066959.

Examples

			a(4) = 13 because 2^2*3^3*4^4 = 2^10*3^3 and 10 + 3 = 13.
		

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000
• Eric Weisstein's World of Mathematics, Hyperfactorial.
• Eric Weisstein's World of Mathematics, K-Function.
• Index entries for sequences computed from exponents in factorization of n.
• Index entries for sequences related to factorial numbers.

Crossrefs

Cf. A001222, A002109, A022559, A066959, A303279.

Programs

Mathematica

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]]
Table[PrimeOmega[Hyperfactorial[n]], {n, 55}]
Table[Sum[k PrimeOmega[k], {k, n}], {n, 55}]
Accumulate[Table[k * PrimeOmega[k], {k, 1, 55}]] (* Amiram Eldar, Jun 13 2025 *)

PARI

a(n) = sum(k=1, n, k*bigomega(k)); \\ Altug Alkan, Apr 20 2018