A106421 - OEIS

%I #11 Sep 12 2024 19:57:34

%S 1,11,10,12,16,108,144,128,1296,1152,1024,10368,10240,12288,16384,

%T 110592,147456,131072,1327104,1179648,1048576,10616832,10485760,

%U 12582912,16777216,113246208,100663296,134217728,1006632960,1207959552

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

%H Robert Israel, <a href="/A106421/b106421.txt">Table of n, a(n) for n = 0..3302</a>

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

%p f:= proc(n) uses priqueue; local pq, t,p,x,i;

%p     initialize(pq);

%p     insert([-2^n,2$n],pq);

%p     do

%p       t:= extract(pq);

%p       x:= -t[1];

%p       if floor(x/10^ilog10(x)) = 1 then return x fi;

%p       p:= nextprime(t[-1]);

%p       for i from n+1 to 2 by -1 while t[i] = t[-1] do

%p         insert([t[1]*(p/t[-1])^(n+2-i), op(t[2..i-1]),p$(n+2-i)],pq)

%p       od;

%p     od

%p end proc:

%p f(0):= 1:

%p map(f, [$0..50]); # _Robert Israel_, Sep 06 2024

%o (Python)

%o from itertools import count

%o from math import isqrt, prod

%o from sympy import primerange, integer_nthroot, primepi

%o def A106421(n):

%o     if n <= 1: return 1+10*n

%o     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     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     for l in count(len(str(1<<n))-1):

%o         kmin, kmax = 10**l-1, 2*10**l-1

%o         mmin, mmax = f(kmin), f(kmax)

%o         if mmax>mmin:

%o             while kmax-kmin > 1:

%o                 kmid = kmax+kmin>>1

%o                 mmid = f(kmid)

%o                 if mmid > mmin:

%o                     kmax, mmax = kmid, mmid

%o                 else:

%o                     kmin, mmin = kmid, mmid

%o     return kmax # _Chai Wah Wu_, Sep 12 2024

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

%K base,nonn

%O 0,2

%A _Ray Chandler_, May 02 2005