A109251 Number of numbers up to 10^n which are products of three primes.

0, 1, 22, 247, 2569, 25556, 250853, 2444359, 23727305, 229924367, 2227121996, 21578747909, 209214982913, 2030133769624, 19717814526785, 191693417109381, 1865380637252270, 18168907486812690, 177123437184971927, 1728190923820610000

Offset: 0

Author

Martin Raab, Aug 19 2005

Keywords

• nonn

Examples

			There are 22 numbers with three prime factors up to 10^2: 8, 12, 18, 20, 27, 28, 30, 42, 44, 45, 50, 52, 63, 66, 68, 70, 75, 76, 78, 92, 98, 99.
		

Crossrefs

Cf. A014612 = numbers with three prime factors, A036352 = number of numbers up to 10^n which are products of two primes, A072114.

Programs

Mathematica

ThreeAlmostPrimePi[n_] := Sum[ PrimePi[n/(Prime@i*Prime@j)] - j + 1, {i, PrimePi[n^(1/3)]}, {j, i, PrimePi@ Sqrt[n/Prime@i]}]; Table[ ThreeAlmostPrimePi[10^n], {n, 0, 14}] (* Robert G. Wilson v *)

Python

from math import isqrt
from sympy import primepi, primerange, integer_nthroot
def A109251(n):
    r = 10**n
    return int(sum(primepi(r//(k*m))-b for a,k in enumerate(primerange(integer_nthroot(r,3)[0]+1)) for b,m in enumerate(primerange(k,isqrt(r//k)+1),a))) # Chai Wah Wu, Sep 18 2024

Formula

a(n) = A072114(10^n). - R. J. Mathar, May 25 2008

Extensions

a(10)-a(14) from Robert G. Wilson v, Feb 06 2006

a(15)-a(17) from Hiroaki Yamanouchi, Aug 30 2014

a(18)-a(19) from Henri Lifchitz, Dec 01 2014