A383413 Area A of triangles such that the sides are distinct integers and A is an integer.

6, 24, 30, 36, 42, 54, 60, 66, 72, 84, 90, 96, 114, 120, 126, 132, 144, 150, 156, 168, 180, 198, 204, 210, 216, 234, 240, 252, 264, 270, 288, 294, 300, 306, 324, 330, 336, 360, 378, 384, 390, 396, 408, 420, 456, 462, 468, 480, 486, 504, 510, 522, 528, 540, 546, 576, 594

Offset: 1

Author

Karl-Heinz Hofmann, Apr 26 2025

Keywords

• nonn

Comments

All terms are multiples of 6.

Subsequence of A188158.

Examples

			72 is in the sequence because the triangle with sides {a=5, b=29, c=30} has an area of exactly 72 and all sides are distinct.
12 is not in the sequence because this area is only possible with the isosceles triangles {a=5, b=5, c=6} and {a=5, b=5, c=8} with a and b not distinct.
		

Links

• Wikipedia, Heronian triangle

Crossrefs

Cf. A188158, A316853.

Programs

Mathematica

nn = 450; lst = {}; Do[s = (a + b + c)/2; If[IntegerQ[s], area2 = s (s - a) (s - b) (s - c); If[a!=b!=c&&0 < area2 <= nn^2 && IntegerQ[Sqrt[area2]], AppendTo[lst, Sqrt[area2]]]], {a, nn}, {b, a-1}, {c, b-1}]; Union[lst]; lst (* James C. McMahon, May 10 2025 *)