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A162143 Numbers that are the squares of the product of three distinct primes.

%I A162143 #31 Apr 14 2025 05:39:40
%S A162143 900,1764,4356,4900,6084,10404,11025,12100,12996,16900,19044,23716,
%T A162143 27225,28900,30276,33124,34596,36100,38025,49284,52900,53361,56644,
%U A162143 60516,65025,66564,70756,74529,79524,81225,81796,84100,96100,101124,103684,119025,125316,127449
%N A162143 Numbers that are the squares of the product of three distinct primes.
%C A162143 Numbers that are the product of exactly 3 distinct squares of primes (p^2*q^2*r^2).
%H A162143 T. D. Noe, <a href="/A162143/b162143.txt">Table of n, a(n) for n = 1..1000</a>
%H A162143 <a href="/index/Pri#prime_signature">Index to sequences related to prime signature</a>
%F A162143 a(n) = A007304(n)^2.
%F A162143 A050326(a(n)) = 8. - _Reinhard Zumkeller_, May 03 2013
%F A162143 Sum_{n>=1} 1/a(n) = (P(2)^3 + 2*P(6) - 3*P(2)*P(4))/6 = (A085548^3 + 2*A085966 - 3*A085548*A085964)/6 = 0.0036962441..., where P is the prime zeta function. - _Amiram Eldar_, Oct 30 2020
%e A162143 900 = 2^2*3^2*5^2, 1764 = 2^2*3^2*7^2, 4356 = 2^2*3^2*11^2, ..
%p A162143 h := proc(n) local P; P := NumberTheory:-PrimeFactors(n); nops(P) = 3 and n = mul(P) end:
%p A162143 A162143List := upto -> seq(n^2, n=select(h, [seq(1..upto)])):  # _Peter Luschny_, Apr 14 2025
%t A162143 fQ[n_]:=Last/@FactorInteger[n]=={2,2,2}; Select[Range[100000], f]
%o A162143 (Python)
%o A162143 from math import isqrt
%o A162143 from sympy import primepi, primerange, integer_nthroot
%o A162143 def A162143(n):
%o A162143     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 A162143     def bisection(f,kmin=0,kmax=1):
%o A162143         while f(kmax) > kmax: kmax <<= 1
%o A162143         while kmax-kmin > 1:
%o A162143             kmid = kmax+kmin>>1
%o A162143             if f(kmid) <= kmid:
%o A162143                 kmax = kmid
%o A162143             else:
%o A162143                 kmin = kmid
%o A162143         return kmax
%o A162143     return bisection(f)**2 # _Chai Wah Wu_, Aug 29 2024
%o A162143 (SageMath)
%o A162143 def is_a(n):
%o A162143     P = prime_divisors(n)
%o A162143     return len(P) == 3 and prod(P) == n
%o A162143 print([n*n for n in range(1, 439) if is_a(n)]) # _Peter Luschny_, Apr 14 2025
%Y A162143 Cf. A007304, A006881, A050326, A162142, A085548, A085964, A085966.
%K A162143 nonn
%O A162143 1,1
%A A162143 _Vladimir Joseph Stephan Orlovsky_, Jun 25 2009
%E A162143 Edited by _N. J. A. Sloane_, Jun 27 2009