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A070136 Numbers m such that [A070080(m), A070081(m), A070082(m)] is a right integer triangle.

%I A070136 #16 Feb 16 2025 08:32:46
%S A070136 17,116,212,370,493,850,1297,1599,1629,2574,2778,3751,4298,4370,5251,
%T A070136 5286,6476,9169,10066,12398,12441,12520,14414,16365,16602,19831,21231,
%U A070136 21486,24060,26125,27245,29230,33625,33658
%N A070136 Numbers m such that [A070080(m), A070081(m), A070082(m)] is a right integer triangle.
%C A070136 Right integer triangles have integer areas: see A070142.
%H A070136 Jean-François Alcover, <a href="/A070136/b070136.txt">Table of n, a(n) for n = 1..137</a>
%H A070136 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/HeronianTriangle.html">Heronian Triangle</a>.
%H A070136 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/RightTriangle.html">Right Triangle</a>.
%H A070136 Reinhard Zumkeller, <a href="/A070080/a070080.txt">Integer-sided triangles</a>
%e A070136 116 is a term: [A070080(116), A070081(116), A070082(116)]=[6,8,10], A070085(116)=6^2+8^2-10^2=36+64-100=0.
%e A070136 212 is a term: [A070080(212), A070081(212), A070082(212)]=[5,12,13], A070085(212)=5^2+12^2-13^2=25+144-169=0.
%t A070136 m = 500 (* max perimeter *);
%t A070136 sides[per_] := Select[Reverse /@ IntegerPartitions[per, {3}, Range[ Ceiling[per/2]]], #[[1]] < per/2 && #[[2]] < per/2 && #[[3]] < per/2 &];
%t A070136 triangles = DeleteCases[Table[sides[per], {per, 3, m}], {}] // Flatten[#, 1]& // SortBy[Total[#] m^3 + #[[1]] m^2 + #[[2]] m + #[[1]] &];
%t A070136 Position[triangles, {a_, b_, c_} /; a^2 + b^2 == c^2] // Flatten (* _Jean-François Alcover_, Oct 12 2021 *)
%Y A070136 Cf. A024155, A070137, A070142, A070085.
%K A070136 nonn
%O A070136 1,1
%A A070136 _Reinhard Zumkeller_, May 05 2002