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A162144 Products of cubes of 3 distinct primes.

%I A162144 #18 Aug 31 2024 01:57:25
%S A162144 27000,74088,287496,343000,474552,1061208,1157625,1331000,1481544,
%T A162144 2197000,2628072,3652264,4492125,4913000,5268024,6028568,6434856,
%U A162144 6859000,7414875,10941048,12167000,12326391,13481272,14886936,16581375,17173512,18821096
%N A162144 Products of cubes of 3 distinct primes.
%C A162144 Numbers of the form p^3*q^3*r^3 where p, q, r are three distinct primes.
%C A162144 The cubic analog of A085986 (squares of 2 distinct primes).
%H A162144 T. D. Noe, <a href="/A162144/b162144.txt">Table of n, a(n) for n = 1..1000</a>
%F A162144 a(n) = (A007304(n))^3.
%F A162144 A000005(a(n)) = 64.
%F A162144 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
%e A162144 27000 = 2^3*3^3*5^3. 74088 = 2^3*3^3*7^3. 287496 = 2^3*3^3*11^3.
%t A162144 fQ[n_]:=Last/@FactorInteger[n]=={1,1,1}; Select[Range[1000], fQ]^3
%o A162144 (Python)
%o A162144 from math import isqrt
%o A162144 from sympy import primepi, primerange, integer_nthroot
%o A162144 def A162144(n):
%o A162144     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)))
%o A162144     def bisection(f,kmin=0,kmax=1):
%o A162144         while f(kmax) > kmax: kmax <<= 1
%o A162144         while kmax-kmin > 1:
%o A162144             kmid = kmax+kmin>>1
%o A162144             if f(kmid) <= kmid:
%o A162144                 kmax = kmid
%o A162144             else:
%o A162144                 kmin = kmid
%o A162144         return kmax
%o A162144     return bisection(f)**3 # _Chai Wah Wu_, Aug 30 2024
%Y A162144 Cf. A006881, A085986, A162142, A162143.
%Y A162144 Cf. A085541, A085966, A085969.
%K A162144 nonn
%O A162144 1,1
%A A162144 _Vladimir Joseph Stephan Orlovsky_, Jun 25 2009
%E A162144 Edited by _R. J. Mathar_, Aug 14 2009