A106427 Smallest number beginning with 7 and having exactly n prime divisors counted with multiplicity.

7, 74, 70, 708, 72, 729, 704, 7056, 768, 7776, 7168, 70656, 71680, 702464, 73728, 746496, 720896, 7225344, 786432, 7962624, 7077888, 71663616, 70778880, 700710912, 75497472, 764411904, 704643072, 7113539584, 7046430720

Offset

1,1

Links

Table of n, a(n) for n=1..29.

Example

a(3) = 70 = 2*5*7.

Prog

(Python)
from itertools import count
from math import isqrt, prod
from sympy import primerange, integer_nthroot, primepi
def A106427(n):
    if n == 1: return 7
    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 = 7*10**l-1, 8*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: A051154 A172257 A367179 * A106417 A137141 A275618

Adjacent sequences: A106424 A106425 A106426 * A106428 A106429 A106430

Keyword

base,nonn

Author

Ray Chandler, May 02 2005

Status

approved