A106429 Smallest number beginning with 9 and having exactly n prime divisors counted with multiplicity.

97, 9, 92, 90, 918, 96, 972, 960, 9072, 9600, 90624, 9216, 93312, 90112, 903168, 98304, 995328, 917504, 9043968, 9175040, 90243072, 9437184, 95551488, 92274688, 924844032, 922746880, 9042919424, 905969664, 9172942848, 9059696640

Offset: 1

Author

Ray Chandler, May 02 2005

Keywords

• base
• nonn

Examples

			a(2) = 9 = 3^2.
		

Crossrefs

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

Programs

PARI

a(n) = {i = 2^n; while ((digits(i)[1] != 9) || (bigomega(i)!=n), i++); i;} \\ Michel Marcus, Sep 14 2013

Python

from itertools import count
from math import isqrt, prod
from sympy import primerange, integer_nthroot, primepi
def A106429(n):
    if n == 1: return 97
    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, Aug 29 2024