A106411 Smallest number beginning with 1 that is the product of exactly n distinct primes.

1, 11, 10, 102, 1110, 10010, 101010, 1009470, 11741730, 1001110110, 10407767370, 1000287585570, 10293281928930, 1001230315195110, 13082761331670030, 1004819888620217670, 100015003602410826930, 1922760350154212639070

Offset: 0

Author

Ray Chandler, May 02 2005

Keywords

• base
• nonn

Examples

			a(0) = 1, a(5) = 10010 = 2*5*7*11*13.
		

Links

• Chai Wah Wu, Table of n, a(n) for n = 0..45

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 A106411(n):
    if n <= 1: return 1+10*n
    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)))
    for l in count(len(str(primorial(n)))-1):
        kmin, kmax = 10**l-1, 2*10**l-1
        mmin, mmax = f(kmin), f(kmax)
        if mmax>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