A116435 Number of n-almost primes less than or equal to n^n.
0, 1, 5, 34, 269, 2613, 28893, 359110, 4934952, 74342563, 1217389949, 21533211312, 409230368646, 8318041706593
Offset: 1
Examples
a(3)=5 because there are five 3-almost primes <= 27, 8,12,18,20&27.
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 *) Do[ Print@ AlmostPrimePi[n, n^n], {n, 13}] -
Python
from math import isqrt, prod from sympy import primerange, integer_nthroot, primepi def A116435(n): def almostprimepi(n,k): 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(n//prod(c[1] for c in a))-a[-1][0] for a in g(n,0,1,1,k)) if k>1 else primepi(n)) return almostprimepi(n**n,n) # Chai Wah Wu, Sep 01 2024
Extensions
a(13)-a(14) from Donovan Johnson, Oct 05 2010
Definition of T(r,c) corrected by R. J. Mathar, Jun 20 2021
Comments