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

1,1

Comments

Right integer triangles have integer areas: see A070143.

Links

Table of n, a(n) for n=1..35.

Eric Weisstein's World of Mathematics, Heronian Triangle.

Eric Weisstein's World of Mathematics, Right Triangle.

Reinhard Zumkeller, Integer-sided triangles

Example

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.

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 *)

Crossrefs

Cf. A070109, A070136.

Sequence in context: A016299 A016250 A255819 * A021054 A016246 A009441

Adjacent sequences: A070134 A070135 A070136 * A070138 A070139 A070140

Keyword

nonn

Author

Reinhard Zumkeller, May 05 2002

Extensions

More terms from Jean-François Alcover, Oct 04 2021

Status

approved