A106418 Smallest number beginning with 8 that is the product of exactly n distinct primes.

83, 82, 805, 858, 8610, 81510, 870870, 80150070, 800509710, 8254436190, 800680310430, 8222980095330, 800160280950030, 80008785365579070, 843685980760953330, 80058789202898516010, 8003887646839494820410

Offset

1,1

Links

Chai Wah Wu, Table of n, a(n) for n = 1..43

Example

a(3) = 805 = 5*7*23.

Prog

(Python)
from itertools import count
from math import prod, isqrt
from sympy import primerange, integer_nthroot, primepi, primorial
def A106418(n):
    if n == 1: return 83
    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)))
    def bisection(f, kmin, kmax, mmin, mmax):
        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
    for l in count(len(str(primorial(n)))-1):
        kmin, kmax = 8*10**l-1, 9*10**l-1
        mmin, mmax = f(kmin), f(kmax)
        if mmax>mmin: return bisection(f, kmin, kmax, mmin, mmax) # Chai Wah Wu, Aug 31 2024

Crossrefs

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

Sequence in context: A134456 A247442 A106428 * A031961 A108751 A256520

Adjacent sequences: A106415 A106416 A106417 * A106419 A106420 A106421

Keyword

base,nonn

Author

Ray Chandler, May 02 2005

Status

approved