A106428 Smallest number beginning with 8 and having exactly n prime divisors counted with multiplicity.

83, 82, 8, 81, 80, 810, 800, 864, 8000, 8064, 80000, 80640, 8192, 82944, 81920, 802816, 819200, 884736, 8126464, 8257536, 80621568, 80216064, 8388608, 84934656, 83886080, 822083584, 838860800, 8120172544, 805306368, 8153726976

Offset: 1

Author

Ray Chandler, May 02 2005

Keywords

• base
• nonn

Examples

			a(3) = 8 = 2^3.
		

Crossrefs

Cf. A077326-A077334, A106411-A106419, A106421-A106429.

Programs

Python

from itertools import count
from math import isqrt, prod
from sympy import primerange, integer_nthroot, primepi
def A106428(n):
    if n == 1: return 83
    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)))
    def f(x): return int(sum(primepi(x//prod(c[1] for c in a))-a[-1][0] for a in g(x,0,1,1,n)))
    for l in count(len(str(1<mmin:
            while kmax-kmin > 1:
                kmid = kmax+kmin>>1
                mmid = f(kmid)
                if mmid > mmin:
                    kmax, mmax = kmid, mmid
                else:
                    kmin, mmin = kmid, mmid
    return kmax # Chai Wah Wu, Sep 12 2024