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A070148 Numbers k such that [A070080(k), A070081(k), A070082(k)] is an integer Heronian triangle having triangular area.

%I A070148 #20 Feb 16 2025 08:32:46
%S A070148 17,368,659,972,1156,1599,1971,2555,2574,3746,3818,4298,4330,5374,
%T A070148 14325,14414,15004,15943,16451,19475,19615,24013,24051,33950,63593,
%U A070148 71630,75052,79286,79670,79921,84183,90187,93290
%N A070148 Numbers k such that [A070080(k), A070081(k), A070082(k)] is an integer Heronian triangle having triangular area.
%H A070148 Jean-François Alcover, <a href="/A070148/b070148.txt">Table of n, a(n) for n = 1..89</a>
%H A070148 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/HeronianTriangle.html">Heronian Triangle</a>.
%H A070148 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RightTriangle.html">Right Triangle</a>.
%H A070148 Reinhard Zumkeller, <a href="/A070080/a070080.txt">Integer-sided triangles</a>
%e A070148 17 is a term: [A070080(17), A070081(17), A070082(17)] = [3,4,5]: A070086(52)=6.
%t A070148 m = 500 (* max perimeter *);
%t A070148 sides[per_] := Select[Reverse /@ IntegerPartitions[per, {3}, Range[ Ceiling[per/2]]], #[[1]] < per/2 && #[[2]] < per/2 && #[[3]] < per/2 &];
%t A070148 triangles = DeleteCases[Table[sides[per], {per, 3, m}], {}] // Flatten[#, 1] & // SortBy[Total[#] m^3 + #[[1]] m^2 + #[[2]] m + #[[1]] &];
%t A070148 area[{a_, b_, c_}] := With[{p = (a + b + c)/2}, Sqrt[p(p-a)(p-b)(p-c)]];
%t A070148 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 A070148 Cf. A070142, A000217.
%K A070148 nonn
%O A070148 1,1
%A A070148 _Reinhard Zumkeller_, May 05 2002