A385819 Numbers k such that there are least five primitive Heron triangles having the same area and perimeter k.

2842, 3542, 5642, 5750, 6314, 7238, 7546, 9790, 15470, 15778, 17710, 20026, 21658, 21970, 22610, 26962

Offset: 1

Author

Zhining Yang, Jul 09 2025

Keywords

• nonn
• hard
• more

Examples

			3542 is a term because there exists 5 primitive Heron triangles: {{421,1518,1603}, {511,1375,1656}, {583,1288,1671},{759,1096,1687}, {851,1001,1690}} with same perimeter 3542 and same area 318780.
20026 is a term because there exists 6 primitive Heron triangles: {{2108,8493,9425}, {2173,8398,9455}, {2261,8277,9488}, {2418,8075,9533}, {4123,6205,9698}, {4588,5729,9709}} with same perimeter 20026 and same area 8410920.
		

Links

• BBS Math Blog, Primitive Heron triangles with equal perimeter and area (in Chinese).

Crossrefs

Cf. A051516, A070138, A188158

Programs

Mathematica

sol = Association[];
For[n = 2, n <= 6000, n += 2,
For[z = Ceiling[n/3], z < Floor[n/2], z++,
For[x = 1, x < Floor[n/3], x++, y = n - x - z;
   If[x + y > z > y > x && GCD[x, y, z] == 1, p = (x + y + z)/2;
    A = Sqrt[p (p - x) (p - y) (p - z)];
    If[IntegerQ[A], d = ToString@n <> "->" <> ToString@A; t = {x, y, z};
     If[KeyExistsQ[sol, d], AppendTo[sol[d], t], sol[d] = {t}]]]]]];
Select[sol, Length@# > 4 &]