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

1,1

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).

Links

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

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

Example

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

Mathematica

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

Prog

(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

Crossrefs

Cf. A006881, A085986, A162142, A162143.

Cf. A085541, A085966, A085969.

Sequence in context: A249496 A227348 A176359 * A305728 A253485 A253480

Adjacent sequences: A162141 A162142 A162143 * A162145 A162146 A162147

Keyword

nonn,changed

Author

Vladimir Joseph Stephan Orlovsky, Jun 25 2009

Extensions

Edited by R. J. Mathar, Aug 14 2009

Status

approved