A106416 - OEIS

%I #11 Sep 27 2024 10:44:58

%S 61,6,66,690,6006,62790,690690,60138078,606996390,6469693230,

%T 600319429710,60007743265470,600277546959090,60039293728424010,

%U 614889782588491410,60865792091025932010,6000526229622444289770

%N Smallest number beginning with 6 that is the product of exactly n distinct primes.

%H Chai Wah Wu, <a href="/A106416/b106416.txt">Table of n, a(n) for n = 1..45</a>

%e a(3) = 66 = 2*3*11.

%o (Python)

%o from itertools import count

%o from math import prod, isqrt

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

%o def A106416(n):

%o     if n == 1: return 61

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

%o         kmin, kmax = 6*10**l-1, 7*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 1,1

%A _Ray Chandler_, May 02 2005