A070137 Numbers k such that [A070080(k), A070081(k), A070082(k)] is a right integer triangle with relatively prime side lengths.

17, 212, 493, 1297, 2574, 4298, 5251, 14414, 16365, 21231, 26125, 39056, 42597, 55042, 63770, 75052, 91121, 97256, 124355, 164640, 200999, 213083, 253721, 275999, 367997, 384154, 415778, 478343, 511633, 518370, 606417, 665040, 689356, 755435, 846571

Offset: 1

Author

Reinhard Zumkeller, May 05 2002

Keywords

• nonn

Comments

Right integer triangles have integer areas: see A070143.

Examples

			493 is a term: [A070080(493), A070081(493), A070082(493)]=[8,15,17], A070084(493)=gcd(8,15,17)=1, A070085(493)=8^2+15^2-17^2=64+225-289=0.
		

Links

• Eric Weisstein's World of Mathematics, Heronian Triangle.
• Eric Weisstein's World of Mathematics, Right Triangle.
• Reinhard Zumkeller, Integer-sided triangles

Crossrefs

Cf. A070109, A070136.

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]] &];
Position[triangles, {a_, b_, c_} /; GCD[a, b, c] == 1 && a^2 + b^2 - c^2 == 0] // Flatten (* Jean-François Alcover, Oct 04 2021 *)

Extensions

More terms from Jean-François Alcover, Oct 04 2021