A360729 a(n) is the number of prime factors of the n-th powerful number (counted with repetition).

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, 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, 7, 8, 2, 9, 6, 7, 2, 6, 8, 7, 6, 11, 4, 7, 3, 2, 10, 5

Offset: 1

Author

Amiram Eldar, Feb 18 2023

Keywords

• nonn

Links

• Amiram Eldar, Table of n, a(n) for n = 1..10000
• Rafael Jakimczuk and Matilde Lalín, The Number of Prime Factors on Average in Certain Integer Sequences, Journal of Integer Sequences, Vol. 25 (2022), Article 22.2.3.

Crossrefs

Cf. A001222, A001694, A083342, A090699.

Similar sequences: A072047, A076399.

Programs

Mathematica

PrimeOmega[Select[Range[3000], # == 1 || Min[FactorInteger[#][[;; , 2]]] > 1 &]]

PARI

apply(bigomega, select(ispowerful, [1..3000]))

Formula

a(n) = A001222(A001694(n)).

Sum_{A001694(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). [corrected Sep 21 2024]