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

Offset

0,3

Links

Table of n, a(n) for n=0..17.

Example

There are 12 primes with four almost primes up to 100: 16,24,36,40,54,56,60,81,84,88,90&100.

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}]

Prog

(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

Crossrefs

Cf. A006880, A066265, A109251, A014613.

Sequence in context: A167138 A001406 A057572 * A015610 A141326 A154733

Adjacent sequences: A114103 A114104 A114105 * A114107 A114108 A114109

Keyword

nonn

Author

Robert G. Wilson v, Feb 07 2006

Extensions

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

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

Status

approved