A106424 Smallest number beginning with 4 and having exactly n prime divisors counted with multiplicity.

41, 4, 42, 40, 48, 400, 432, 4000, 4032, 40000, 4608, 4096, 41472, 40960, 49152, 409600, 442368, 4063232, 4128768, 40310784, 4718592, 4194304, 42467328, 41943040, 411041792, 419430400, 452984832, 402653184, 4076863488, 4026531840

Offset

1,1

Links

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

Example

a(1) = 41, a(3) = 42 = 2*3*7.

Prog

(Python)
from itertools import count
from math import isqrt, prod
from sympy import primerange, integer_nthroot, primepi
def A106424(n):
    if n == 1: return 41
    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 = 4*10**l-1, 5*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: A051317 A036189 A198190 * A126636 A095188 A215540

Adjacent sequences: A106421 A106422 A106423 * A106425 A106426 A106427

Keyword

base,nonn

Author

Ray Chandler, May 02 2005

Status

approved