A106418 Smallest number beginning with 8 that is the product of exactly n distinct primes.

83, 82, 805, 858, 8610, 81510, 870870, 80150070, 800509710, 8254436190, 800680310430, 8222980095330, 800160280950030, 80008785365579070, 843685980760953330, 80058789202898516010, 8003887646839494820410

Offset: 1

Author

Ray Chandler, May 02 2005

Keywords

• base
• nonn

Examples

			a(3) = 805 = 5*7*23.
		

Links

• Chai Wah Wu, Table of n, a(n) for n = 1..43

Crossrefs

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

Programs

Python

from itertools import count
from math import prod, isqrt
from sympy import primerange, integer_nthroot, primepi, primorial
def A106418(n):
    if n == 1: return 83
    def g(x,a,b,c,m): yield from (((d,) for d in enumerate(primerange(b+1,isqrt(x//c)+1),a+1)) if m==2 else (((a2,b2),)+d for a2,b2 in enumerate(primerange(b+1,integer_nthroot(x//c,m)[0]+1),a+1) 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)))
    def bisection(f,kmin,kmax,mmin,mmax):
        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
    for l in count(len(str(primorial(n)))-1):
        kmin, kmax = 8*10**l-1, 9*10**l-1
        mmin, mmax = f(kmin), f(kmax)
        if mmax>mmin: return bisection(f,kmin,kmax,mmin,mmax) # Chai Wah Wu, Aug 31 2024