A120053 Number of 12-almost primes less than or equal to 10^n.

0, 0, 0, 0, 3, 63, 865, 11068, 133862, 1563465, 17836903, 200051717, 2214357712, 24255601105, 263439785143, 2841076717752, 30457549169277, 324855769153426, 3449587218984911, 36489283363168885

Offset

0,5

Links

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

Example

There are 3 twelve-almost primes up to 10000: 4096, 6144, and 9216.

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[12, 10^n], {n, 11}]

Prog

(Python)
from math import prod, isqrt
from sympy import primerange, integer_nthroot, primepi
def A120053(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, 12))) # 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: A048354 A046181 A151993 * A139293 A370827 A133275

Adjacent sequences: A120050 A120051 A120052 * A120054 A120055 A120056

Keyword

nonn,more

Author

Robert G. Wilson v, Feb 07 2006

Extensions

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

a(15) from Chai Wah Wu, Aug 24 2024

a(16)-a(19) from Henri Lifchitz, Mar 18 2025

Status

approved