A106417 Smallest number beginning with 7 that is the product of exactly n distinct primes.

7, 74, 70, 714, 7410, 71610, 746130, 70387590, 703600590, 70015935990, 700288518930, 7420738134810, 701098433345310, 70016268785853390, 757887406446280110, 70025936403159126390, 7001749954335151685670

Offset

1,1

Links

Chai Wah Wu, Table of n, a(n) for n = 1..45

Example

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

Prog

(Python)
from itertools import count
from math import prod, isqrt
from sympy import primerange, integer_nthroot, primepi, primorial
def A106417(n):
    if n == 1: return 7
    def g(x, a, b, c, m): yield from (((d, ) for d in enumerate(primerange(b+1, isqrt(x//c)+1), a+1)) if m==2 else (((a2, b2), )+d for a2, b2 in enumerate(primerange(b+1, integer_nthroot(x//c, m)[0]+1), a+1) 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(primorial(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: A172257 A367179 A106427 * A137141 A275618 A114472

Adjacent sequences: A106414 A106415 A106416 * A106418 A106419 A106420

Keyword

base,nonn

Author

Ray Chandler, May 02 2005

Status

approved