login
A106426
Smallest number beginning with 6 and having exactly n prime divisors counted with multiplicity.
2
61, 6, 63, 60, 612, 64, 648, 640, 6048, 6400, 6912, 6144, 62208, 61440, 602112, 65536, 663552, 655360, 6029312, 6553600, 60162048, 6291456, 63700992, 62914560, 616562688, 67108864, 679477248, 603979776, 6115295232, 6039797760
OFFSET
1,1
EXAMPLE
a(4) = 60 = 2^2*3*5.
PROG
(Python)
from itertools import count
from math import isqrt, prod
from sympy import primerange, integer_nthroot, primepi
def A106426(n):
if n == 1: return 61
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 = 6*10**l-1, 7*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