A120047 Number of 6-almost primes less than or equal to 10^n.

0, 0, 2, 37, 485, 5933, 68963, 774078, 8493366, 91683887, 977694273, 10327249593, 108264085934, 1128049914377, 11694704489580, 120734708167792, 1242063105505230, 12739510126065301, 130330025583399801

Offset

0,3

Links

Table of n, a(n) for n=0..18.

Example

There are 2 six-almost primes up to 100: 64 and 96, so a(2) = 2.

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[6, 10^n], {n, 0, 13}]

Prog

(Python)
from math import isqrt, prod
from sympy import primerange, integer_nthroot, primepi
def almostprimepi(n, k):
    if k==0: return int(n>=1)
    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(n//prod(c[1] for c in a))-a[-1][0] for a in g(n, 0, 1, 1, k)) if k>1 else primepi(n))
def A120047(n): return almostprimepi(10**n, 6) # Chai Wah Wu, Dec 09 2024

Crossrefs

Cf. A066265, A014613, A116382.

Cf. A006880, A036352, A109251, A114106, A114453.

Cf. A120048, A120049, A120050, A120051, A120052, A120053, A116430.

Sequence in context: A300542 A100849 A300969 * A300926 A337306 A216539

Adjacent sequences: A120044 A120045 A120046 * A120048 A120049 A120050

Keyword

nonn,more,changed

Author

Robert G. Wilson v, Feb 07 2006

Extensions

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

a(15)-a(18) from Henri Lifchitz, Feb 03 2025

Status

approved