A116426 The number of n-almost primes less than or equal to 4^n, starting with a(0)=1.

1, 2, 6, 13, 34, 77, 177, 406, 887, 1962, 4225, 9094, 19482, 41414, 87706, 184976, 389357, 816193, 1708412, 3566209, 7431153, 15457234, 32098652, 66560309, 137830562, 285062028, 588871107, 1215176367, 2505048537, 5159228725

Offset: 0

Author

Robert G. Wilson v, Feb 10 2006

Keywords

• nonn

Crossrefs

Cf. A078840, A078841, A078842, A116432, A078843, A116426, A078844, A116427, A078845, A116428, A116429, A116430, A078846, A116431.

Programs

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 *)
Join[{1},Table[AlmostPrimePi[n, 4^n], {n, 29}]]

Python

from math import isqrt, prod
from sympy import primerange, integer_nthroot, primepi
def A116426(n):
    if n<=1: return 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((1<<(n<<1))//prod(c[1] for c in a))-a[-1][0] for a in g(1<<(n<<1),0,1,1,n))) # Chai Wah Wu, Oct 02 2024