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A106428
Smallest number beginning with 8 and having exactly n prime divisors counted with multiplicity.
2
83, 82, 8, 81, 80, 810, 800, 864, 8000, 8064, 80000, 80640, 8192, 82944, 81920, 802816, 819200, 884736, 8126464, 8257536, 80621568, 80216064, 8388608, 84934656, 83886080, 822083584, 838860800, 8120172544, 805306368, 8153726976
OFFSET
1,1
EXAMPLE
a(3) = 8 = 2^3.
PROG
(Python)
from itertools import count
from math import isqrt, prod
from sympy import primerange, integer_nthroot, primepi
def A106428(n):
if n == 1: return 83
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)))
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)))
for l in count(len(str(1<<n))-1):
kmin, kmax = 8*10**l-1, 9*10**l-1
mmin, mmax = f(kmin), f(kmax)
if mmax>mmin:
while kmax-kmin > 1:
kmid = kmax+kmin>>1
mmid = f(kmid)
if mmid > mmin:
kmax, mmax = kmid, mmid
else:
kmin, mmin = kmid, mmid
return kmax # Chai Wah Wu, Sep 12 2024
KEYWORD
base,nonn
AUTHOR
Ray Chandler, May 02 2005
STATUS
approved