A096468 Perimeters of primitive Heronian triangles.

12, 16, 18, 30, 32, 36, 40, 42, 44, 48, 50, 54, 56, 60, 64, 66, 68, 70, 72, 76, 78, 80, 84, 90, 96, 98, 100, 104, 108, 110, 112, 114, 120, 126, 128, 130, 132, 136, 140, 144, 150, 152, 154, 156, 160, 162, 164, 168, 170, 172, 174, 176, 180, 182, 186, 190, 192, 196

Offset: 1

Author

T. D. Noe, Jun 22 2004

Keywords

• nonn

Comments

Here a primitive Heronian triangle has integer sides a,b,c with GCD(a,b,c) = 1 and integral area. The perimeter is always even. Cheney's article contains many theorems about these triangles.

Examples

			12 is on this list because the triangle with sides 3, 4, 5 has integral area and perimeter 12.
		

Links

• Peter Kagey and Giovanni Resta, Table of n, a(n) for n = 1..10000 (first 1000 terms from Peter Kagey)
• Wm. Fitch Cheney, Jr., Heronian Triangles, Amer. Math. Monthly, Vol. 36, No. 1 (Jan 1929), 22-28.
• Eric Weisstein's World of Mathematics, Heronian Triangle

Crossrefs

Cf. A070138 (number of primitive Heronian triangles having perimeter n), A083875 (area/6 of primitive Heronian triangles), A096467 (longest side of primitive Heronian triangles).

Programs

Mathematica

nn=150; lst={}; Do[s=(a+b+c)/2; If[IntegerQ[s] && GCD[a, b, c]==1, area2=s(s-a)(s-b)(s-c); If[area2>0 && IntegerQ[Sqrt[area2]], AppendTo[lst, 2s]]], {a, nn}, {b, a}, {c, b}]; Union[lst]

Extensions

Name changed by Wesley Ivan Hurt, May 17 2020