A303279 Expansion of (1/(1 - x)^2) * Sum_{p prime, k>=1} x^(p^k)/(1 - x^(p^k)).

0, 1, 3, 7, 12, 19, 27, 38, 51, 66, 82, 101, 121, 143, 167, 195, 224, 256, 289, 325, 363, 403, 444, 489, 536, 585, 637, 692, 748, 807, 867, 932, 999, 1068, 1139, 1214, 1290, 1368, 1448, 1532, 1617, 1705, 1794, 1886, 1981, 2078, 2176, 2279, 2384, 2492, 2602, 2715, 2829, 2947, 3067

Offset

1,3

Comments

Sum of exponents in prime-power factorization of product of first n factorials (A000178).

Partial sums of A022559.

Links

Daniel Suteu, Table of n, a(n) for n = 1..10000

Eric Weisstein's World of Mathematics, Barnes G-Function

Eric Weisstein's World of Mathematics, Superfactorial

Index entries for sequences related to factorial numbers

Index entries for sequences computed from exponents in factorization of n

Example

a(5) = 12 because 2!*3!*4!*5! = 2^8*3^3*5 and 8 + 3 + 1 = 12.

Maple

b:= proc(n) option remember; `if`(n<1, [0$2],
      (p-> p+[numtheory[bigomega](n), p[1]])(b(n-1)))
    end:
a:= n-> b(n+1)[2]:
seq(a(n), n=1..55);  # Alois P. Heinz, Oct 07 2021

Mathematica

nmax = 55; Rest[CoefficientList[Series[1/(1 - x)^2 Sum[Boole[PrimePowerQ[k]] x^k/(1 - x^k), {k, 1, nmax}], {x, 0, nmax}], x]]
Table[PrimeOmega[BarnesG[n + 2]], {n, 55}]
Table[ Sum[ PrimeOmega@ j, {k, n}, {j, k}], {n, 55}]

Prog

(PARI) a(n) = my(t=0); sum(k=1, n, t+=bigomega(k)); \\ Daniel Suteu, Jan 17 2019

Crossrefs

Cf. A000178, A001222, A022559, A303281.

Sequence in context: A194147 A077043 A022330 * A024219 A371701 A283733

Adjacent sequences: A303276 A303277 A303278 * A303280 A303281 A303282

Keyword

nonn

Author

Ilya Gutkovskiy, Apr 20 2018

Status

approved