OFFSET
0,2
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[n, 12^n], {n, 12}]
PROG
(PARI)
almost_prime_count(N, k) = if(k==1, return(primepi(N))); (f(m, p, k, j=0) = my(c=0, s=sqrtnint(N\m, k)); if(k==2, forprime(q=p, s, c += primepi(N\(m*q))-j; j += 1), forprime(q=p, s, c += f(m*q, q, k-1, j); j += 1)); c); f(1, 2, k);
a(n) = if(n == 0, 1, almost_prime_count(12^n, n)); \\ Daniel Suteu, Jul 10 2023
(Python)
from math import isqrt, prod
from sympy import primerange, integer_nthroot, primepi
def A116431(n):
if n<=1: return 4*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(12**n//prod(c[1] for c in a))-a[-1][0] for a in g(12**n, 0, 1, 1, n))) # Chai Wah Wu, Sep 28 2024
CROSSREFS
KEYWORD
nonn,more
AUTHOR
Robert G. Wilson v, Feb 10 2006
EXTENSIONS
a(13)-a(14) from Donovan Johnson, Oct 01 2010
a(15)-a(16) from Daniel Suteu, Jul 10 2023
STATUS
approved