A106421 Smallest number beginning with 1 and having exactly n prime divisors counted with multiplicity.

1, 11, 10, 12, 16, 108, 144, 128, 1296, 1152, 1024, 10368, 10240, 12288, 16384, 110592, 147456, 131072, 1327104, 1179648, 1048576, 10616832, 10485760, 12582912, 16777216, 113246208, 100663296, 134217728, 1006632960, 1207959552

Offset: 0

Author

Ray Chandler, May 02 2005

Keywords

• base
• nonn

Examples

			a(0) = 1, a(5) = 108 = 2^2*3^3.
		

Links

• Robert Israel, Table of n, a(n) for n = 0..3302

Crossrefs

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

Programs

Maple

f:= proc(n) uses priqueue; local pq, t,p,x,i;
    initialize(pq);
    insert([-2^n,2$n],pq);
    do
      t:= extract(pq);
      x:= -t[1];
      if floor(x/10^ilog10(x)) = 1 then return x fi;
      p:= nextprime(t[-1]);
      for i from n+1 to 2 by -1 while t[i] = t[-1] do
        insert([t[1]*(p/t[-1])^(n+2-i), op(t[2..i-1]),p$(n+2-i)],pq)
      od;
    od
end proc:
f(0):= 1:
map(f, [$0..50]); # Robert Israel, Sep 06 2024

Python

from itertools import count
from math import isqrt, prod
from sympy import primerange, integer_nthroot, primepi
def A106421(n):
    if n <= 1: return 1+10*n
    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