A123073 Number of ordered triples of primes (p,q,r) such that pqr = n-th 3-almost prime A014612(n).

1, 3, 3, 3, 1, 3, 6, 6, 3, 3, 3, 3, 3, 6, 3, 6, 3, 3, 6, 3, 3, 3, 6, 6, 6, 6, 3, 3, 3, 1, 6, 6, 3, 3, 3, 6, 3, 6, 6, 3, 3, 6, 3, 6, 6, 3, 6, 6, 3, 3, 6, 6, 6, 3, 6, 3, 3, 3, 6, 6, 6, 3, 6, 3, 6, 3, 3, 6, 3, 6, 6, 6, 3, 6, 3, 6, 6, 3, 3, 3, 3, 1, 6, 6, 3, 6, 3, 6, 3, 6, 6, 6, 3, 3, 6, 6, 3, 6, 6, 3, 6, 3, 3, 6, 3

Offset: 1

Author

N. J. A. Sloane and T. D. Noe, Sep 29 2006

Keywords

• nonn

Comments

The nonzero subsequence of A123074.

Crossrefs

Cf. A123074, A014612.

Programs

Python

from math import isqrt
from sympy import primepi, primerange, integer_nthroot, primefactors
def A123073(n):
    def bisection(f,kmin=0,kmax=1):
        while f(kmax) > kmax: kmax <<= 1
        while kmax-kmin > 1:
            kmid = kmax+kmin>>1
            if f(kmid) <= kmid:
                kmax = kmid
            else:
                kmin = kmid
        return kmax
    def f(x): return int(n+x-sum(primepi(x//(k*m))-b for a,k in enumerate(primerange(integer_nthroot(x,3)[0]+1)) for b,m in enumerate(primerange(k,isqrt(x//k)+1),a)))
    return (1,3,6)[len(primefactors(bisection(f,n,n)))-1] # Chai Wah Wu, Oct 20 2024

Extensions

More terms from T. D. Noe, Sep 29 2006