A072047 Number of prime factors of the squarefree numbers: omega(A005117(n)).

0, 1, 1, 1, 2, 1, 2, 1, 1, 2, 2, 1, 1, 2, 2, 1, 2, 1, 3, 1, 2, 2, 2, 1, 2, 2, 1, 3, 1, 2, 1, 2, 1, 2, 2, 2, 1, 1, 2, 2, 3, 1, 2, 3, 1, 1, 2, 2, 3, 1, 2, 1, 2, 2, 2, 1, 2, 2, 2, 2, 1, 1, 3, 1, 3, 2, 1, 1, 3, 2, 1, 3, 2, 2, 2, 2, 2, 1, 2, 3, 1, 2, 2, 1, 3, 1, 2

Offset: 1

Author

Reinhard Zumkeller, Jun 09 2002

Keywords

• nonn
• nice

Comments

For n > 1: length of row n in A265668. - Reinhard Zumkeller, Dec 13 2015

Links

• Reinhard Zumkeller, 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. A072048.

Cf. A001221, A001222, A013661, A005117, A265668.

Programs

Haskell

a072047 n = a072047_list !! (n-1)
a072047_list = map a001221 $ a005117_list
-- Reinhard Zumkeller, Aug 08 2011

Mathematica

PrimeOmega[Select[Range[200],SquareFreeQ]] (* Harvey P. Dale, May 14 2011 *)

PARI

apply(omega, select(issquarefree, [1..200])) \\ Michel Marcus, Nov 25 2022

Python

from math import isqrt
from sympy import mobius, primenu
def A072047(n):
    def f(x): return n+x-sum(mobius(k)*(x//k**2) for k in range(1, isqrt(x)+1))
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    return primenu(bisection(f)) # Chai Wah Wu, Aug 31 2024

Formula

a(n) = A001221(A005117(n)) = A001222(A005117(n)).

Sum_{A005117(k) <= x} a(k) = (1/zeta(2))*x*log(log(x)) + O(x) (Jakimczuk and Lalín, 2022). - Amiram Eldar, Feb 18 2023, corrected Sep 21 2024