

A096468


Perimeters of primitive Heronian triangles.


36



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
(list;
graph;
refs;
listen;
history;
text;
internal format)



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

Wm. Fitch Cheney, Jr., Heronian Triangles, Amer. Math. Monthly, Vol. 36, No. 1 (Jan 1929), 2228.


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(sa)(sb)(sc); 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).


KEYWORD

nonn


AUTHOR



EXTENSIONS



STATUS

approved



