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

%I A303279 #45 Aug 02 2025 06:52:21
%S A303279 0,1,3,7,12,19,27,38,51,66,82,101,121,143,167,195,224,256,289,325,363,
%T A303279 403,444,489,536,585,637,692,748,807,867,932,999,1068,1139,1214,1290,
%U A303279 1368,1448,1532,1617,1705,1794,1886,1981,2078,2176,2279,2384,2492,2602,2715,2829,2947,3067
%N A303279 Expansion of (1/(1 - x)^2) * Sum_{p prime, k>=1} x^(p^k)/(1 - x^(p^k)).
%C A303279 Sum of exponents in prime-power factorization of product of first n factorials (A000178).
%C A303279 Partial sums of A022559.
%H A303279 Daniel Suteu, <a href="/A303279/b303279.txt">Table of n, a(n) for n = 1..10000</a>
%H A303279 Jean-Marie De Koninck and William Verreault, <a href="https://doi.org/10.2298/PIM2429045D">Arithmetic functions at factorial arguments</a>, Publications de l'Institut Mathématique, Vol. 115, No. 129 (2024), pp. 45-76.
%H A303279 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/BarnesG-Function.html">Barnes G-Function</a>.
%H A303279 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/Superfactorial.html">Superfactorial</a>.
%H A303279 <a href="/index/Eu#epf">Index entries for sequences computed from exponents in factorization of n</a>.
%H A303279 <a href="/index/Fa#factorial">Index entries for sequences related to factorial numbers</a>.
%F A303279 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
%F A303279 a(n) = (1/2) * Sum_{k=0..n} A356718(n,k). - _Dario T. de Castro_, Feb 19 2025
%e A303279 a(5) = 12 because 2!*3!*4!*5! = 2^8*3^3*5 and 8 + 3 + 1 = 12.
%p A303279 b:= proc(n) option remember; `if`(n<1, [0$2],
%p A303279       (p-> p+[numtheory[bigomega](n), p[1]])(b(n-1)))
%p A303279     end:
%p A303279 a:= n-> b(n+1)[2]:
%p A303279 seq(a(n), n=1..55);  # _Alois P. Heinz_, Oct 07 2021
%t A303279 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]]
%t A303279 Table[PrimeOmega[BarnesG[n + 2]], {n, 55}]
%t A303279 Table[ Sum[ PrimeOmega@ j, {k, n}, {j, k}], {n, 55}]
%o A303279 (PARI) a(n) = my(t=0); sum(k=1, n, t+=bigomega(k)); \\ _Daniel Suteu_, Jan 17 2019
%Y A303279 Cf. A000178, A001222, A022559, A083342, A303281.
%Y A303279 Half of row sums of triangle A356718.
%K A303279 nonn
%O A303279 1,3
%A A303279 _Ilya Gutkovskiy_, Apr 20 2018