A106418 - OEIS

%I #12 Sep 12 2024 19:35:13

%S 83,82,805,858,8610,81510,870870,80150070,800509710,8254436190,

%T 800680310430,8222980095330,800160280950030,80008785365579070,

%U 843685980760953330,80058789202898516010,8003887646839494820410

%N Smallest number beginning with 8 that is the product of exactly n distinct primes.

%H Chai Wah Wu, <a href="/A106418/b106418.txt">Table of n, a(n) for n = 1..43</a>

%e a(3) = 805 = 5*7*23.

%o (Python)

%o from itertools import count

%o from math import prod, isqrt

%o from sympy import primerange, integer_nthroot, primepi, primorial

%o def A106418(n):

%o     if n == 1: return 83

%o     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)))

%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     def bisection(f,kmin,kmax,mmin,mmax):

%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

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

%o         kmin, kmax = 8*10**l-1, 9*10**l-1

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

%o         if mmax>mmin: return bisection(f,kmin,kmax,mmin,mmax) # _Chai Wah Wu_, Aug 31 2024

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

%K base,nonn

%O 1,1

%A _Ray Chandler_, May 02 2005