A106422 Smallest number beginning with 2 and having exactly n prime divisors counted with multiplicity.

2, 21, 20, 24, 200, 216, 288, 256, 2592, 2304, 2048, 20736, 20480, 24576, 204800, 221184, 294912, 262144, 2654208, 2359296, 2097152, 21233664, 20971520, 25165824, 209715200, 226492416, 201326592, 268435456, 2013265920, 2415919104

Offset

1,1

Links

Table of n, a(n) for n=1..30.

Example

a(1) = 2, a(5) = 200 = 2^3*5^2.

Prog

(Python)
from itertools import count
from math import isqrt, prod
from sympy import primerange, integer_nthroot, primepi
def A106422(n):
    if n == 1: return 2
    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 = 2*10**l-1, 3*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

Crossrefs

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

Sequence in context: A212483 A084313 A140268 * A320523 A371060 A171549

Adjacent sequences: A106419 A106420 A106421 * A106423 A106424 A106425

Keyword

base,nonn

Author

Ray Chandler, May 02 2005

Status

approved