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%I A106423 #11 Sep 12 2024 19:57:45
%S A106423 3,33,30,36,32,324,320,384,3200,3456,3072,31104,30720,36864,32768,
%T A106423 331776,327680,393216,3276800,3538944,3145728,31850496,31457280,
%U A106423 37748736,33554432,339738624,301989888,3057647616,3019898880,3623878656
%N A106423 Smallest number beginning with 3 and having exactly n prime divisors counted with multiplicity.
%H A106423 Robert Israel, <a href="/A106423/b106423.txt">Table of n, a(n) for n = 1..3303</a>
%e A106423 a(1) = 3, a(6) = 324 = 2^2*3^4.
%p A106423 f:= proc(n) uses priqueue; local pq, t,p,x,i;
%p A106423     initialize(pq);
%p A106423     insert([-2^n,2$n],pq);
%p A106423     do
%p A106423       t:= extract(pq);
%p A106423       x:= -t[1];
%p A106423       if floor(x/10^ilog10(x)) = 3 then return x fi;
%p A106423       p:= nextprime(t[-1]);
%p A106423       for i from n+1 to 2 by -1 while t[i] = t[-1] do
%p A106423         insert([t[1]*(p/t[-1])^(n+2-i), op(t[2..i-1]),p$(n+2-i)],pq)
%p A106423       od;
%p A106423     od
%p A106423 end proc:
%p A106423 map(f, [$1..50]); # _Robert Israel_, Sep 06 2024
%o A106423 (Python)
%o A106423 from itertools import count
%o A106423 from math import isqrt, prod
%o A106423 from sympy import primerange, integer_nthroot, primepi
%o A106423 def A106423(n):
%o A106423     if n == 1: return 3
%o A106423     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)))
%o A106423     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)))
%o A106423     for l in count(len(str(1<<n))-1):
%o A106423         kmin, kmax = 3*10**l-1, 4*10**l-1
%o A106423         mmin, mmax = f(kmin), f(kmax)
%o A106423         if mmax>mmin:
%o A106423             while kmax-kmin > 1:
%o A106423                 kmid = kmax+kmin>>1
%o A106423                 mmid = f(kmid)
%o A106423                 if mmid > mmin:
%o A106423                     kmax, mmax = kmid, mmid
%o A106423                 else:
%o A106423                     kmin, mmin = kmid, mmid
%o A106423     return kmax # _Chai Wah Wu_, Sep 12 2024
%Y A106423 Cf. A077326-A077334, A106411-A106419, A106421-A106429.
%K A106423 base,nonn
%O A106423 1,1
%A A106423 _Ray Chandler_, May 02 2005

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