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

1,2

Links

Table of n, a(n) for n=1..19.

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

Example

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

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 *)

Prog

(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

Crossrefs

Cf. A007304.

Sequence in context: A085506 A307084 A132508 * A159355 A184577 A229772

Adjacent sequences: A215215 A215216 A215217 * A215219 A215220 A215221

Keyword

nonn

Author

Martin Renner, Aug 06 2012

Extensions

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

Status

approved