A162144 - OEIS

%I #18 Aug 31 2024 01:57:25

%S 27000,74088,287496,343000,474552,1061208,1157625,1331000,1481544,

%T 2197000,2628072,3652264,4492125,4913000,5268024,6028568,6434856,

%U 6859000,7414875,10941048,12167000,12326391,13481272,14886936,16581375,17173512,18821096

%N Products of cubes of 3 distinct primes.

%C Numbers of the form p^3*q^3*r^3 where p, q, r are three distinct primes.

%C The cubic analog of A085986 (squares of 2 distinct primes).

%H T. D. Noe, <a href="/A162144/b162144.txt">Table of n, a(n) for n = 1..1000</a>

%F a(n) = (A007304(n))^3.

%F A000005(a(n)) = 64.

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

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

%o (Python)

%o from math import isqrt

%o from sympy import primepi, primerange, integer_nthroot

%o def A162144(n):

%o     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     def bisection(f,kmin=0,kmax=1):

%o         while f(kmax) > kmax: kmax <<= 1

%o         while kmax-kmin > 1:

%o             kmid = kmax+kmin>>1

%o             if f(kmid) <= kmid:

%o                 kmax = kmid

%o             else:

%o                 kmin = kmid

%o         return kmax

%o     return bisection(f)**3 # _Chai Wah Wu_, Aug 30 2024

%Y Cf. A006881, A085986, A162142, A162143.

%Y Cf. A085541, A085966, A085969.

%K nonn

%O 1,1

%A _Vladimir Joseph Stephan Orlovsky_, Jun 25 2009

%E Edited by _R. J. Mathar_, Aug 14 2009