A162144 Products of cubes of 3 distinct primes.

27000, 74088, 287496, 343000, 474552, 1061208, 1157625, 1331000, 1481544, 2197000, 2628072, 3652264, 4492125, 4913000, 5268024, 6028568, 6434856, 6859000, 7414875, 10941048, 12167000, 12326391, 13481272, 14886936, 16581375, 17173512, 18821096

Offset: 1

Author

Vladimir Joseph Stephan Orlovsky, Jun 25 2009

Keywords

• nonn

Comments

Numbers of the form p^3*q^3*r^3 where p, q, r are three distinct primes.

The cubic analog of A085986 (squares of 2 distinct primes).

Examples

			27000 = 2^3*3^3*5^3. 74088 = 2^3*3^3*7^3. 287496 = 2^3*3^3*11^3.
		

Links

• T. D. Noe, Table of n, a(n) for n = 1..1000

Crossrefs

Cf. A006881, A085986, A162142, A162143.

Cf. A085541, A085966, A085969.

Programs

Mathematica

fQ[n_]:=Last/@FactorInteger[n]=={1,1,1}; Select[Range[1000], fQ]^3

Python

from math import isqrt
from sympy import primepi, primerange, integer_nthroot
def A162144(n):
    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),1) for b,m in enumerate(primerange(k+1,isqrt(x//k)+1),a+1)))
    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
    return bisection(f)**3 # Chai Wah Wu, Aug 30 2024

Formula

a(n) = (A007304(n))^3.

A000005(a(n)) = 64.

Sum_{n>=1} 1/a(n) = (P(3)^3 + 2*P(9) - 3*P(3)*P(6))/6 = (A085541^3 + 2*A085969 - 3*A085541*A085966)/6 = 0.0000661486..., where P is the prime zeta function. - Amiram Eldar, Oct 30 2020

Extensions

Edited by R. J. Mathar, Aug 14 2009