|
%I #15 May 12 2023 15:29:12
%S 0,2,3,2,4,2,3,5,4,2,6,5,4,4,5,2,3,7,6,2,4,5,6,4,5,8,7,2,6,3,2,5,6,7,
%T 4,4,5,9,2,8,4,7,5,4,6,6,7,2,8,6,2,5,7,6,10,4,5,9,4,4,8,5,3,5,2,5,4,4,
%U 7,8,2,9,6,7,2,6,8,7,6,11,4,7,3,2,10,5
%N a(n) is the number of prime factors of the n-th powerful number (counted with repetition).
%H Amiram Eldar, <a href="/A360729/b360729.txt">Table of n, a(n) for n = 1..10000</a>
%H Rafael Jakimczuk and Matilde Lalín, <a href="https://cs.uwaterloo.ca/journals/JIS/VOL25/Lalin/lalin2.html">The Number of Prime Factors on Average in Certain Integer Sequences</a>, Journal of Integer Sequences, Vol. 25 (2022), Article 22.2.3.
%F a(n) = A001222(A001694(n)).
%F Sum_{a(k) < x} a(k) = (2*zeta(3/2)/zeta(3))*sqrt(x)*log(log(x)) + (2*(B_2 - log(2)) + Sum_{p prime} (3/((p^(3/2)+1))))*(zeta(3/2)/zeta(3))*sqrt(x) + O(sqrt(x)/sqrt(log(x))), where B_2 = A083342 (Jakimczuk and Lalín, 2022).
%t PrimeOmega[Select[Range[3000], # == 1 || Min[FactorInteger[#][[;; , 2]]] > 1 &]]
%o (PARI) apply(bigomega, select(ispowerful, [1..3000]))
%Y Cf. A001222, A001694, A083342, A090699.
%Y Similar sequences: A072047, A076399.
%K nonn
%O 1,2
%A _Amiram Eldar_, Feb 18 2023
|
