A120049 Number of 8-almost primes less than or equal to 10^n.

0, 0, 0, 7, 105, 1418, 17572, 207207, 2367507, 26483012, 291646797, 3173159326, 34192782745, 365561221293, 3882841742380

Offset

0,4

Links

Table of n, a(n) for n=0..14.

Example

There are 7 eight-almost primes up to 1000: 256, 384, 576, 640, 864, 896 & 960.

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[8, 10^n], {n, 12}]

Prog

(Python)
from math import prod, isqrt
from sympy import primerange, integer_nthroot, primepi
def A120049(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, 8))) # Chai Wah Wu, Aug 23 2024

Crossrefs

Cf. A066265, A014613, A116382.

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

Sequence in context: A098362 A093741 A139742 * A067420 A368351 A358957

Adjacent sequences: A120046 A120047 A120048 * A120050 A120051 A120052

Keyword

nonn

Author

Robert G. Wilson v, Feb 07 2006

Extensions

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

Example corrected by Harvey P. Dale, Aug 13 2018

Status

approved