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
Keywords
Examples
a(5) = 12 because 2!*3!*4!*5! = 2^8*3^3*5 and 8 + 3 + 1 = 12.
Links
- Daniel Suteu, Table of n, a(n) for n = 1..10000
- Jean-Marie De Koninck and William Verreault, Arithmetic functions at factorial arguments, Publications de l'Institut Mathématique, Vol. 115, No. 129 (2024), pp. 45-76.
- Eric Weisstein's World of Mathematics, Barnes G-Function.
- Eric Weisstein's World of Mathematics, Superfactorial.
- Index entries for sequences computed from exponents in factorization of n.
- Index entries for sequences related to factorial numbers.
Programs
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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}] -
PARI
a(n) = my(t=0); sum(k=1, n, t+=bigomega(k)); \\ Daniel Suteu, Jan 17 2019
Formula
a(n) = (n^2/2) * (log(log(n)) + c + O(1/log(n))), where c = A083342 (De Koninck and Verreault, 2024, p. 49, eq. (3.1)). - Amiram Eldar, Dec 10 2024
a(n) = (1/2) * Sum_{k=0..n} A356718(n,k). - Dario T. de Castro, Feb 19 2025
Comments