A162142 Numbers that are the cube of a product of two distinct primes (p^3*q^3).

216, 1000, 2744, 3375, 9261, 10648, 17576, 35937, 39304, 42875, 54872, 59319, 97336, 132651, 166375, 185193, 195112, 238328, 274625, 328509, 405224, 456533, 551368, 614125, 636056, 658503, 753571, 804357, 830584, 857375, 1191016, 1367631, 1520875, 1643032

Offset: 1

Author

Vladimir Joseph Stephan Orlovsky, Jun 25 2009

Keywords

• nonn

Comments

Subset of A046306, of A000578, and of A007774. - R. J. Mathar, Jun 27 2009

Examples

			216=2^3*3^3. 1000=2^3*5^3. 2744=2^3*7^3.
		

Links

• T. D. Noe, Table of n, a(n) for n = 1..1000
• Index to sequences related to prime signature

Crossrefs

Cf. A000578, A006881, A007774, A046306.

Cf. A085541, A085966.

Programs

Mathematica

fQ[n_]:=Last/@FactorInteger[n]=={3,3}; lst={};Do[If[fQ[n],AppendTo[lst, n]],{n,6*9!}];lst
With[{nn=30},Select[Union[(Times@@@Subsets[Prime[Range[nn]],{2}])^3],#<= (2Prime[ nn])^3&]](* Harvey P. Dale, May 27 2024 *)

Python

from math import isqrt
from sympy import primepi, primerange
def A162142(n):
    def f(x): return int(n+x+(t:=primepi(s:=isqrt(x)))+(t*(t-1)>>1)-sum(primepi(x//k) for k in primerange(1, s+1)))
    m, k = n, f(n)
    while m != k: m, k = k, f(k)
    return m**3 # Chai Wah Wu, Dec 09 2024

Formula

a(n) = (A006881(n))^3 = A000578(A006881(n)). - R. J. Mathar, Jun 27 2009

Sum_{n>=1} 1/a(n) = (P(3)^2 - P(6))/2 = (A085541^2 - A085966)/2 = 0.006735..., where P is the prime zeta function. - Amiram Eldar, Jul 06 2020

Extensions

Definition rephrased by R. J. Mathar, Jun 27 2009