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%I #19 Oct 06 2021 13:36:56
%S 17,368,659,972,1156,1599,1971,2555,2574,3746,3818,4298,4330,5374,
%T 14325,14414,15004,15943,16451,19475,19615,24013,24051,33950,63593,
%U 71630,75052,79286,79670,79921,84183,90187,93290
%N Numbers k such that [A070080(k), A070081(k), A070082(k)] is an integer Heronian triangle having triangular area.
%H Jean-François Alcover, <a href="/A070148/b070148.txt">Table of n, a(n) for n = 1..89</a>
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/HeronianTriangle.html">Heronian Triangle</a>.
%H Eric Weisstein's World of Mathematics, <a href="http://mathworld.wolfram.com/RightTriangle.html">Right Triangle</a>.
%H Reinhard Zumkeller, <a href="/A070080/a070080.txt">Integer-sided triangles</a>
%e 17 is a term: [A070080(17), A070081(17), A070082(17)] = [3,4,5]: A070086(52)=6.
%t m = 500 (* max perimeter *);
%t sides[per_] := Select[Reverse /@ IntegerPartitions[per, {3}, Range[ Ceiling[per/2]]], #[[1]] < per/2 && #[[2]] < per/2 && #[[3]] < per/2 &];
%t triangles = DeleteCases[Table[sides[per], {per, 3, m}], {}] // Flatten[#, 1] & // SortBy[Total[#] m^3 + #[[1]] m^2 + #[[2]] m + #[[1]] &];
%t area[{a_, b_, c_}] := With[{p = (a + b + c)/2}, Sqrt[p(p-a)(p-b)(p-c)]];
%t Position[triangles, {a_, b_, c_} /; IntegerQ[area[{a, b, c}]] && IntegerQ[Sqrt[1 + 8 area[{a, b, c}]]]] // Flatten (* _Jean-François Alcover_, Oct 04 2021 *)
%Y Cf. A070142, A000217.
%K nonn
%O 1,1
%A _Reinhard Zumkeller_, May 05 2002
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