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

1,1

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

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

Example

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.

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 *)

Crossrefs

Cf. A120131, A120132, A120133, A224301.

Sequence in context: A356863 A112548 A032620 * A096468 A334983 A334984

Adjacent sequences: A224299 A224300 A224301 * A224303 A224304 A224305

Keyword

nonn

Author

Mihir Mathur, Apr 04 2013

Extensions

Corrected and extended by Giovanni Resta, Apr 04 2013

Status

approved