A116431 The number of n-almost primes less than or equal to 12^n, starting with a(0)=1.
1, 5, 48, 434, 3695, 29165, 218283, 1569995, 10950776, 74621972, 499495257, 3297443264, 21533211312, 139411685398, 896352197825, 5730605551626, 36465861350230
Offset: 0
Crossrefs
Programs
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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}] -
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
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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
Extensions
a(13)-a(14) from Donovan Johnson, Oct 01 2010
a(15)-a(16) from Daniel Suteu, Jul 10 2023