A106413 Smallest number beginning with 3 that is the product of exactly n distinct primes.

3, 33, 30, 330, 3570, 30030, 3015870, 30120090, 300690390, 30043474230, 304075581810, 30035662366710, 304250263527210, 30078810535603830, 3001252188252588270, 32589158477190044730, 3003056284355533696290

Offset: 1

Author

Ray Chandler, May 02 2005

Keywords

• base
• nonn

Examples

			a(1) = 3, a(6) = 30030 = 2*3*5*7*11*13.
		

Links

• Chai Wah Wu, Table of n, a(n) for n = 1..45

Crossrefs

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

Programs

Maple

f:= proc(n) uses priqueue; local pq, t, p, x, i,L,v,Lp;
    initialize(pq);
    L:= [seq(ithprime(i),i=1..n)];
    v:= convert(L,`*`);
    insert([-v, L], pq);
    do
      t:= extract(pq);
      x:= -t[1];
      if floor(x/10^ilog10(x)) = 3 then return x fi;
      L:= t[2];
      p:= nextprime(L[-1]);
      for i from n to 1 by -1 do
        if i < n and L[i] <> prevprime(L[i+1]) then break fi;
        Lp:= [op(L[1..i-1]),op(L[i+1..n]),p];
        insert([-convert(Lp,`*`),Lp], pq)
    od od;
end proc:
map(f, [$1..30]); # Robert Israel, Sep 12 2024

Python

from itertools import count
from math import prod, isqrt
from sympy import primerange, integer_nthroot, primepi, primorial
def A106413(n):
    if n == 1: return 3
    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 = 3*10**l-1, 4*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