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A120053
Number of 12-almost primes less than or equal to 10^n.
10
0, 0, 0, 0, 3, 63, 865, 11068, 133862, 1563465, 17836903, 200051717, 2214357712, 24255601105, 263439785143, 2841076717752
OFFSET
0,5
EXAMPLE
There are 3 twelve-almost primes up to 10000: 4096, 6144 & 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
KEYWORD
nonn
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
STATUS
approved