A215218 Number of sphenic numbers, i.e., numbers with exactly three distinct prime factors, up to 10^n.

0, 5, 135, 1800, 19919, 206964, 2086746, 20710806, 203834084, 1997171674, 19522428788, 190614467420, 1860310801454, 18155356377267, 177224592578839, 1730651760050923, 16908343191198752, 165279853754232019, 1616504757072680964

Offset: 1

Author

Martin Renner, Aug 06 2012

Keywords

• nonn

Examples

			a(2) = 5 since there are the five sphenic numbers 30, 42, 66, 70, 78 up to 100.
		

Links

• Paul Kinlaw, Lower bounds for numbers with three prime factors, Husson University, Bangor, ME, 2019. Also in Integers (2019) 19, Article #A22.

Crossrefs

Cf. A007304.

Programs

Mathematica

f[n_] := Sum[ PrimePi[n/(Prime@ i*Prime@ j)] - j, {i, PrimePi[n^(1/3)]}, {j, i +1, PrimePi@ Sqrt[n/Prime@ i]}]; (* Robert G. Wilson v, Dec 28 2016 *)

Python

from math import isqrt
from sympy import primepi, primerange, integer_nthroot
def A215218(n): return int(sum(primepi(10**n//(k*m))-b for a,k in enumerate(primerange(integer_nthroot(10**n,3)[0]+1),1) for b,m in enumerate(primerange(k+1,isqrt(10**n//k)+1),a+1))) # Chai Wah Wu, Aug 26 2024

Extensions

a(8)-a(19) from Henri Lifchitz, Nov 11 2012