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

1,1

Comments

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

Links

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

Index to sequences related to prime signature

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

Example

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

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 *)

Prog

(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

Crossrefs

Cf. A000578, A006881, A007774, A046306.

Cf. A085541, A085966.

Sequence in context: A111029 A124581 A177493 * A233859 A233853 A245993

Adjacent sequences: A162139 A162140 A162141 * A162143 A162144 A162145

Keyword

nonn

Author

Vladimir Joseph Stephan Orlovsky, Jun 25 2009

Extensions

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

Status

approved