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

1,1

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.

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

Example

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

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]

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).

Sequence in context: A112548 A032620 A224302 * A334983 A334984 A334338

Adjacent sequences: A096465 A096466 A096467 * A096469 A096470 A096471

Keyword

nonn

Author

T. D. Noe, Jun 22 2004

Extensions

Name changed by Wesley Ivan Hurt, May 17 2020

Status

approved