A224301 Sorted areas of primitive integer Heronian triangles.

6, 12, 12, 24, 30, 36, 36, 42, 60, 60, 60, 60, 66, 72, 84, 84, 84, 84, 90, 90, 114, 120, 120, 120, 126, 126, 126, 132, 156, 156, 168, 168, 168, 180, 180, 198, 204, 210, 210, 210, 210, 210, 210, 216, 234, 240, 252, 252, 252, 264, 264, 270, 288, 300, 300, 306

Offset: 1

Author

Mihir Mathur, Apr 03 2013

Keywords

• nonn

Comments

The sequence gives the sorted areas of primitive triangles which have integer side lengths and integer areas.

Examples

			The smallest Heronian triangle is (3,4,5) as perimeter and area are integers. The first term of the sequence is thus the area of this triangle, which is 6.
		

Links

• Giovanni Resta, Table of n, a(n) for n = 1..10000
• Michael Somos, Heronian Triangle Table

Crossrefs

Cf. A120131, A120132, A120133, A072294, A188158.

Programs

Mathematica

AMax=400;
Do[
  c=p/b;
  a1=Sqrt[b^2+c^2+2Sqrt[b^2c^2-4A^2]];
  a2=Sqrt[b^2+c^2-2Sqrt[b^2c^2-4A^2]];
  If[IntegerQ[a2]&&GCD[a2,b,c]==1&&a1>a2>=b,A//Sow(*{A,a2,b,c}//Sow*)];
  If[IntegerQ[a1]&&GCD[a1,b,c]==1,A//Sow(*{A,a1,b,c}//Sow*)];
  ,{A,6,AMax,6}
  ,{p,4A^2//Divisors//Select[#,EvenQ[#]&&#>=2A&]&//#/2+2A^2/#&//Select[#,IntegerQ]&}
  ,{b,p//Divisors//Select[#,#^2>=p&]&}
]//Reap//Last//Last
{a1,a2,c}=.;
(* Albert Lau, May 20 2016 *)

Extensions

Definition corrected by Giovanni Resta, Apr 03 2013

More terms from Giovanni Resta, Apr 03 2013