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

%I A162142 #30 Dec 09 2024 23:23:46
%S A162142 216,1000,2744,3375,9261,10648,17576,35937,39304,42875,54872,59319,
%T A162142 97336,132651,166375,185193,195112,238328,274625,328509,405224,456533,
%U A162142 551368,614125,636056,658503,753571,804357,830584,857375,1191016,1367631,1520875,1643032
%N A162142 Numbers that are the cube of a product of two distinct primes (p^3*q^3).
%C A162142 Subset of A046306, of A000578, and of A007774. - _R. J. Mathar_, Jun 27 2009
%H A162142 T. D. Noe, <a href="/A162142/b162142.txt">Table of n, a(n) for n = 1..1000</a>
%H A162142 <a href="/index/Pri#prime_signature">Index to sequences related to prime signature</a>
%F A162142 a(n) = (A006881(n))^3 = A000578(A006881(n)). - _R. J. Mathar_, Jun 27 2009
%F A162142 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
%e A162142 216=2^3*3^3. 1000=2^3*5^3. 2744=2^3*7^3.
%t A162142 fQ[n_]:=Last/@FactorInteger[n]=={3,3}; lst={};Do[If[fQ[n],AppendTo[lst, n]],{n,6*9!}];lst
%t A162142 With[{nn=30},Select[Union[(Times@@@Subsets[Prime[Range[nn]],{2}])^3],#<= (2Prime[ nn])^3&]](* _Harvey P. Dale_, May 27 2024 *)
%o A162142 (Python)
%o A162142 from math import isqrt
%o A162142 from sympy import primepi, primerange
%o A162142 def A162142(n):
%o A162142     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)))
%o A162142     m, k = n, f(n)
%o A162142     while m != k: m, k = k, f(k)
%o A162142     return m**3 # _Chai Wah Wu_, Dec 09 2024
%Y A162142 Cf. A000578, A006881, A007774, A046306.
%Y A162142 Cf. A085541, A085966.
%K A162142 nonn
%O A162142 1,1
%A A162142 _Vladimir Joseph Stephan Orlovsky_, Jun 25 2009
%E A162142 Definition rephrased by _R. J. Mathar_, Jun 27 2009