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A363052 Integers m for which there exist positive integers j, k such that j*k*(j+k) = m^2.

%I A363052 #27 Jul 03 2023 12:20:37
%S A363052 4,18,24,32,36,50,60,108,140,144,150,192,252,256,288,300,360,392,400,
%T A363052 480,486,500,540,588,648,780,816,864,882,900,972,1008,1014,1050,1120,
%U A363052 1152,1156,1176,1200,1350,1372,1452,1536,1620,1764,1800,1848,2016,2040,2048,2178
%N A363052 Integers m for which there exist positive integers j, k such that j*k*(j+k) = m^2.
%C A363052 All terms are even.
%e A363052 24 is a term: j*k*(j+k) = 24^2 for j=2, k=16.
%t A363052 Select[2*Range@500,
%t A363052  Length@Select[Table[(Sqrt[b^2 + 4 #^2/b] - b)/2, {b, #}], IntegerQ] >
%t A363052     0 &]
%t A363052 Select[Union@
%t A363052   Flatten@Table[Sqrt[a*b (a + b)], {a, 1, 80}, {b, a, 500}],
%t A363052  IntegerQ[#] && # < 1000 &]
%o A363052 (Python)
%o A363052 from itertools import count, islice
%o A363052 from sympy import integer_nthroot, divisors
%o A363052 def A363052_gen(startvalue=1): # generator of terms >= startvalue
%o A363052     for m in count(max(startvalue,1)):
%o A363052         for k in divisors(m**2,generator=True):
%o A363052             p, q = integer_nthroot(k**4+(k*m**2<<2),2)
%o A363052             if q:
%o A363052                 a, b = divmod(p-k**2,k<<1)
%o A363052                 if a > 0 and not b:
%o A363052                     yield m
%o A363052                     break
%o A363052 A363052_list = list(islice(A363052_gen(),20)) # _Chai Wah Wu_, Jul 03 2023
%Y A363052 Cf. A088915.
%K A363052 nonn,easy
%O A363052 1,1
%A A363052 _Zhining Yang_, May 15 2023