A289173 The largest n-almost prime less than 3^n.

2, 6, 20, 60, 208, 624, 2080, 6240, 18720, 58240, 176000, 529408, 1593344, 4780032, 14344192, 43040768, 129138688, 387416064, 1162248192, 3486777344, 10460332032, 31380996096, 94142988288, 282428964864, 847286894592, 2541860683776, 7625582051328

Offset: 1

Author

Zak Seidov, Jun 26 2017

Keywords

• nonn

Comments

All terms are even as 3^n is the first odd n-almost prime.

Examples

			a(26) = 2541860683776 = 3^26 - 5144553 = 2^18*3^6*47*283 (a 26-almost prime).
From _Michael De Vlieger_, Jun 27 2017: (Start)
Table of prime factors of a(n) for 1 <= n <= 16:
   1:  2
   2:  2   3
   3:  2   2   5
   4:  2   2   3   5
   5:  2   2   2   2  13
   6:  2   2   2   2   3  13
   7:  2   2   2   2   2   5  13
   8:  2   2   2   2   2   3   5  13
   9:  2   2   2   2   2   3   3   5  13
  10:  2   2   2   2   2   2   2   5   7  13
  11:  2   2   2   2   2   2   2   5   5   5  11
  12:  2   2   2   2   2   2   2   2   2   2  11  47
  13:  2   2   2   2   2   2   2   2   2   2   2   2 389
  14:  2   2   2   2   2   2   2   2   2   2   2   2   3 389
  15:  2   2   2   2   2   2   2   2   2   2   2   2   2  17 103
  16:  2   2   2   2   2   2   2   2   2   2   2   2   2   2  37  71(End)
		

Links

• Jon E. Schoenfield, Table of n, a(n) for n = 1..100

Crossrefs

Cf. A078843 (where 3^n occurs in n-almost primes).

Programs

Mathematica

Table[SelectFirst[Range[3^n - 1, 2^n, -1], PrimeOmega@ # == n &], {n, 18}] (* Michael De Vlieger, Jun 27 2017 *)

PARI

for (n = 1,26, m = 3^n-1; while(bigomega(m) <> n, m = m-2); print1 (m ","))

PARI

a(n)=my(target=n-1); forstep(k=3^n\2,1,-1, if(bigomega(k)==target, return(2*k))) \\ Charles R Greathouse IV, Jul 05 2017

Python

from math import prod, isqrt
from sympy import primepi, primerange, integer_nthroot
def A289173(n):
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        kmin = kmax >> 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    def almostprimepi(n,k):
        if k==0: return int(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(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))
    m = almostprimepi(3**n-1,n)
    def f(x): return m+x-almostprimepi(x,n)
    return bisection(f,m,m) # Chai Wah Wu, Mar 29 2025

Extensions

a(27) from Jon E. Schoenfield, Jul 02 2017