A224302 Sorted perimeters of primitive Heronian triangles.

12, 16, 18, 30, 32, 36, 36, 40, 42, 42, 44, 48, 50, 50, 54, 54, 54, 56, 60, 64, 64, 64, 66, 68, 70, 70, 72, 76, 78, 80, 80, 84, 84, 84, 84, 84, 90, 90, 90, 96, 98, 98, 98, 98, 98, 98, 100, 100, 100, 104, 104, 108, 108, 108, 108, 108, 110, 112, 112, 112, 112

Offset: 1

Author

Mihir Mathur, Apr 04 2013

Keywords

• nonn

Comments

Here a primitive Heronian triangle has integer sides a,b,c with gcd(a,b,c) = 1 and integral area.

The perimeters of primitive Heronian triangles are even [Wenzel Šimerka, 1869]. - Mo Li, Feb 02 2020

Examples

			a(1) = 12 as it is the perimeter of the Heronian triangle having sides 3,4,5 and is the smallest Heronian triangle perimeter.
a(2) = 16 as it is the perimeter of the Heronian triangle having sides 5,5,6 and is the 2nd smallest Heronian triangle perimeter.
		

References

• L. E. Dickson, History of the Theory of Numbers, vol. II: Diophantine Analysis, Dover, 2005, p. 196. [21a]

Links

• Giovanni Resta, Table of n, a(n) for n = 1..10002
• L. E. Dickson, History of the Theory of Numbers, vol. II, 1952, see p. 196 [21 a].
• Michael Somos, Heronian Triangle Table
• Wikipedia, Heronian triangle

Crossrefs

Cf. A120131, A120132, A120133, A224301.

Programs

Mathematica

hQ[a_, b_, c_] := IntegerQ@Sqrt@Block[{s = (a + b + c)/2}, s (s - a) (s - b) (s - c)];
Sort[Reap[Do[If[GCD[a, b, c] == 1 && hQ[a, b, c], Sow@(a + b + c)], {a, 100}, {b, a}, {c, a - b + 1, b}]][[2, 1]]] (* The last numbers given may not be exactly in the right place. *) (* Jinyuan Wang, Feb 02 2020 *)

Extensions

Corrected and extended by Giovanni Resta, Apr 04 2013