A078843 Where 3^n occurs in n-almost primes, starting at a(0)=1.

1, 2, 3, 5, 8, 14, 23, 39, 64, 103, 169, 269, 427, 676, 1065, 1669, 2628, 4104, 6414, 10023, 15608, 24281, 37733, 58503, 90616, 140187, 216625, 334527, 516126, 795632, 1225641, 1886570, 2901796, 4460359, 6851532, 10518476, 16138642, 24748319

Offset: 0

Author

Benoit Cloitre and Paul D. Hanna, Dec 10 2002

Keywords

• nonn

Examples

			a(3) = 5 since 3^3 is the 5th 3-almost-prime: 8,12,18,20,27,....., A014612.
		

Links

• Max Alekseyev, Table of n, a(n) for n = 0..50
• Eric Weisstein's World of Mathematics, Almost Prime.

Crossrefs

Cf. A078840, A078841, A078842, A078844, A078845, A078846.

Programs

Mathematica

AlmostPrimePi[k_Integer /; k > 1, n_] := Module[{a, i}, a[0] = 1; 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, 3^n], {n, 2, 37}] (* Robert G. Wilson v, Feb 09 2006 *)

PARI

a(n)=sum(i=1,3^n,if(bigomega(i)-n,0,1))

PARI

{ appi(k,n,m=2) = local(r=0);
if(k==0,return(1));
if(k==1,return(primepi(n)));
forprime(p=m, floor(sqrtn(n,k)+1e-20),
r+=appi(k-1,n\p,p)-(k==2)*(primepi(p)-1));
r }
{ appi3(k,n) = appi(k,n) - if(k>=1,appi(k-1,n\3)) }
a=1; for(n=1,50, k=ceil(n*log(5/3)/log(5/2)); a+=appi3(n-k,3^n\2^k); print1(a,", "))
\\ Max Alekseyev, Jan 06 2008

Python

from math import isqrt, prod
from sympy import primerange, integer_nthroot, primepi
def A078843(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(3**n,n) if n else 1 # Chai Wah Wu, Sep 01 2024

Formula

a(n) = a(n-1) + appi3(n-k, floor(3^n/2^k)), where k = ceiling(n*c) with c = log(5/3)/log(5/2) = 0.55749295065024006729857073190835923443... and appi3(k,n) is the number of k-almost primes not divisible by 3 and not exceeding n. - Max Alekseyev, Jan 06 2008

Extensions

a(14)-a(37) from Robert G. Wilson v, Feb 09 2006