A120050 Number of 9-almost primes less than or equal to 10^n.

0, 0, 0, 2, 47, 671, 8491, 101787, 1180751, 13377156, 148930536, 1636170477, 17787688377, 191742524399, 2052389350029, 21838745177567, 231206458686127, 2437121982958248, 25591920108631224, 267840642082525459

Offset

0,4

Links

Table of n, a(n) for n=0..19.

Example

There are 2 nine-almost primes up to 1000: 512 & 768.

Mathematica

AlmostPrimePi[k_Integer, n_] := Module[{a, i}, a[0] = 1; If[k == 1, PrimePi[n], Sum[PrimePi[n/Times @@ Prime[Array[a, k - 1]]] - a[k - 1] + 1, Evaluate[ Sequence @@ Table[{a[i], a[i - 1], PrimePi[(n/Times @@ Prime[Array[a, i - 1]])^(1/(k - i + 1))]}, {i, k - 1}]]]]]; (* Eric W. Weisstein, Feb 07 2006 *)
Table[AlmostPrimePi[9, 10^n], {n, 12}]

Prog

(Python)
from math import isqrt, prod
from sympy import primerange, integer_nthroot, primepi
def A120050(n):
    def g(x, a, b, c, m): yield from (((d, ) for d in enumerate(primerange(b, isqrt(x//c)+1), a)) if m==2 else (((a2, b2), )+d for a2, b2 in enumerate(primerange(b, integer_nthroot(x//c, m)[0]+1), a) for d in g(x, a2, b2, c*b2, m-1)))
    return int(sum(primepi(10**n//prod(c[1] for c in a))-a[-1][0] for a in g(10**n, 0, 1, 1, 9))) # Chai Wah Wu, Oct 20 2025

Crossrefs

Cf. A066265, A014613, A116382.

Cf. A006880, A036352, A109251, A114106, A114453, A120047, A120048, A120049, A120050, A120051, A120052, A120053, A116430.

Sequence in context: A226420 A160922 A276509 * A112783 A277655 A246543

Adjacent sequences: A120047 A120048 A120049 * A120051 A120052 A120053

Keyword

nonn

Author

Robert G. Wilson v, Feb 07 2006

Extensions

a(13)-a(14) from Robert G. Wilson v, Jan 07 2007

a(15)-a(19) from Henri Lifchitz, Mar 18 2025

Status

approved