A114106 Number of 4-almost primes less than or equal to 10^n.

0, 0, 12, 149, 1712, 18744, 198062, 2050696, 20959322, 212385942, 2139236881, 21454599814, 214499908019, 2139634739326, 21306682904040, 211905511283590, 2105504493045818, 20905484578206982, 207458897819329031, 2057931819347474462

Offset: 0

Author

Robert G. Wilson v, Feb 07 2006

Keywords

• nonn

Examples

			There are 12 primes with four almost primes up to 100: 16, 24, 36, 40, 54, 56, 60, 81, 84, 88, 90 and 100, so a(2) = 12.
		

Crossrefs

Cf. A006880, A066265, A109251, A014613.

Programs

Mathematica

FourAlmostPrimePi[n_] := Sum[ PrimePi[n/(Prime@i*Prime@j*Prime@k)] - k + 1, {i, PrimePi[n^(1/4)]}, {j, i, PrimePi[(n/Prime@i)^(1/3)]}, {k, j, PrimePi@Sqrt[n/(Prime@i*Prime@j)]}]; Table[ FourAlmostPrimePi[n], {n, 0, 13}]

Python

from math import isqrt
from sympy import primepi, primerange, integer_nthroot
def A114106(n): return sum(primepi(10**n//(k*m*r))-c for a,k in enumerate(primerange(integer_nthroot(10**n,4)[0]+1)) for b,m in enumerate(primerange(k,integer_nthroot(10**n//k,3)[0]+1),a) for c,r in enumerate(primerange(m,isqrt(10**n//(k*m))+1),b)) # Chai Wah Wu, Aug 17 2024

Extensions

a(14) from Robert G. Wilson v, Jan 07 2007

a(15)-a(17) from Henri Lifchitz, Jul 21 2015

a(18)-a(19) from Henri Lifchitz, Feb 02 2025