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

1,2

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.

Formula

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

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).

Mathematica

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

Prog

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

Crossrefs

Cf. A001222, A001694, A083342, A090699.

Similar sequences: A072047, A076399.

Sequence in context: A025477 A080189 A076399 * A370328 A286602 A286600

Adjacent sequences: A360726 A360727 A360728 * A360730 A360731 A360732

Keyword

nonn

Author

Amiram Eldar, Feb 18 2023

Status

approved