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A299170 List of integer triples (b,c,d) where b > c > d are coprime and 1/b^2 + 1/c^2 + 1/d^2 = 1/r^2 and r is an integer, ordered by b then c.

%I A299170 #86 Mar 04 2018 15:55:12
%S A299170 156,65,45,156,80,65,255,136,90,255,160,136,609,580,315,609,580,560,
%T A299170 1295,444,315,1295,560,444,1428,221,91,1560,1547,170,1640,369,270,
%U A299170 1640,480,369,1833,884,799,1924,663,629,2385,1484,945,2385,1680,1484,2925,1100,429
%N A299170 List of integer triples (b,c,d) where b > c > d are coprime and 1/b^2 + 1/c^2 + 1/d^2 = 1/r^2 and r is an integer, ordered by b then c.
%C A299170 Conjectures:
%C A299170 12|r, 3|b or 3|c or 3|d, 4|b or 4|c or 4|d.
%C A299170 No term is powerful (A001694) or square (A000290).
%H A299170 Giovanni Resta, <a href="/A299170/b299170.txt">Table of n, a(n) for n = 1..570</a>
%F A299170 a(n) > 1.
%e A299170 1/156^2 + 1/65^2 + 1/45^2 = 1/36^2 = 1/(12*3)^2.
%e A299170 As an array, sequence begins:
%e A299170    156,   65,   45
%e A299170    156,   80,   65,
%e A299170    255,  136,   90,
%e A299170    255,  160,  136,
%e A299170    609,  580,  315,
%e A299170    609,  580,  560,
%e A299170   1295,  444,  315,
%e A299170   1295,  560,  444,
%e A299170   1428,  221,   91,
%e A299170   1560, 1547,  170,
%e A299170   1640,  369,  270,
%e A299170   1640,  480,  369,
%e A299170   1833,  884,  799,
%e A299170   1924,  663,  629,
%e A299170   ...
%t A299170 n = 1500; lst = {}; Do[Do[Do[If[GCD[b, c, d] == 1,
%t A299170 r = Sqrt[1/(1/b^2 + 1/c^2 + 1/d^2)];
%t A299170   If[IntegerQ[r], lst = AppendTo[lst, {b, c, d}]]], {d, c - 1}],
%t A299170 {c, b - 1}], {b, n}]; lst//Flatten
%Y A299170 Cf. A065607, A120692, A120693.
%K A299170 nonn,tabf
%O A299170 1,1
%A A299170 _Ralf Steiner_, Feb 04 2018
%E A299170 a(28)-a(51) from _Giovanni Resta_, Feb 06 2018