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

17, 368, 659, 972, 1156, 1599, 1971, 2555, 2574, 3746, 3818, 4298, 4330, 5374, 14325, 14414, 15004, 15943, 16451, 19475, 19615, 24013, 24051, 33950, 63593, 71630, 75052, 79286, 79670, 79921, 84183, 90187, 93290

Offset: 1

Author

Reinhard Zumkeller, May 05 2002

Keywords

• nonn

Examples

			17 is a term: [A070080(17), A070081(17), A070082(17)] = [3,4,5]: A070086(52)=6.
		

Links

• Jean-François Alcover, Table of n, a(n) for n = 1..89
• Eric Weisstein's World of Mathematics, Heronian Triangle.
• Eric Weisstein's World of Mathematics, Right Triangle.
• Reinhard Zumkeller, Integer-sided triangles

Crossrefs

Cf. A070142, A000217.

Programs

Mathematica

m = 500 (* max perimeter *);
sides[per_] := Select[Reverse /@ IntegerPartitions[per, {3}, Range[ Ceiling[per/2]]], #[[1]] < per/2 && #[[2]] < per/2 && #[[3]] < per/2 &];
triangles = DeleteCases[Table[sides[per], {per, 3, m}], {}] // Flatten[#, 1] & // SortBy[Total[#] m^3 + #[[1]] m^2 + #[[2]] m + #[[1]] &];
area[{a_, b_, c_}] := With[{p = (a + b + c)/2}, Sqrt[p(p-a)(p-b)(p-c)]];
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 *)