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

1,1

Comments

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

Links

Giovanni Resta, Table of n, a(n) for n = 1..10000

Michael Somos, Heronian Triangle Table

Example

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.

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 *)

Crossrefs

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

Sequence in context: A070149 A315597 A315598 * A181798 A323332 A330922

Adjacent sequences: A224298 A224299 A224300 * A224302 A224303 A224304

Keyword

nonn

Author

Mihir Mathur, Apr 03 2013

Extensions

Definition corrected by Giovanni Resta, Apr 03 2013

More terms from Giovanni Resta, Apr 03 2013

Status

approved