A106429 - OEIS

%I #11 Aug 30 2024 02:54:45

%S 97,9,92,90,918,96,972,960,9072,9600,90624,9216,93312,90112,903168,

%T 98304,995328,917504,9043968,9175040,90243072,9437184,95551488,

%U 92274688,924844032,922746880,9042919424,905969664,9172942848,9059696640

%N Smallest number beginning with 9 and having exactly n prime divisors counted with multiplicity.

%e a(2) = 9 = 3^2.

%o (PARI) a(n) = {i = 2^n; while ((digits(i)[1] != 9) || (bigomega(i)!=n), i++); i;} \\ _Michel Marcus_, Sep 14 2013

%o (Python)

%o from itertools import count

%o from math import isqrt, prod

%o from sympy import primerange, integer_nthroot, primepi

%o def A106429(n):

%o     if n == 1: return 97

%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 = 9*10**l-1, 10**(l+1)-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_, Aug 29 2024

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

%K base,nonn

%O 1,1

%A _Ray Chandler_, May 02 2005