A343769 Sorted areas of primitive Heronian triangles for which a rectangle exists with integer dimensions and with perimeter and area equal respectively to the perimeter and area of the triangle.

12, 126, 624, 1260, 1800, 2100, 2850, 4536, 5292, 5580, 8820, 9900, 12600, 12642, 14850, 15600, 17640, 19110, 21756, 23400, 24948, 25200, 25536, 28350, 47040, 47304

Offset: 1

Author

Jason Zimba, Apr 28 2021

Keywords

• nonn
• more

Examples

			a(1) = 12 because 12 is the area of the 5-5-6 triangle, which is the least-area primitive Heronian triangle for which a rectangle exists with integer dimensions (2-by-6) and with perimeter (16) and area (12) equal respectively to the perimeter and area of the triangle.
a(2) = 126 because 126 is the area of the 13-20-21 triangle, which is the second-least-area primitive Heronian triangle for which a rectangle exists with integer dimensions (6-by-21) and with perimeter (54) and area (126) equal respectively to the perimeter and area of the triangle.
		

Links

• Jason Zimba, There are infinitely many rectangular Heronian triangles.

Crossrefs

Subsequence of A224301.

Programs

Mathematica

(* Adapted from Albert Lau's program for A224301 *)
AMax = 10000;
Do[c = p/b;
    a1 = Sqrt[b^2 + c^2 + 2 Sqrt[b^2 c^2 - 4 A^2]];
    a2 = Sqrt[b^2 + c^2 - 2 Sqrt[b^2 c^2 - 4 A^2]];
    If[IntegerQ[a2] && GCD[a2, b, c] == 1 &&
      a1 > a2 >= b && (per = a2 + b + c;
       IntegerQ[(per + Sqrt[per^2 - 16 A])/4]), A // Sow(*{A,a2,b,c}//
     Sow*)];
    If[IntegerQ[a1] &&
      GCD[a1, b, c] == 1 && (per = a1 + b + c;
       IntegerQ[(per + Sqrt[per^2 - 16 A])/4]), A // Sow(*{A,a1,b,c}//
     Sow*)];, {A, 6, AMax, 6}, {p,
     4 A^2 // Divisors //
        Select[#, EvenQ[#] && # >= 2 A &] & // #/2 + 2 A^2/# & //
      Select[#, IntegerQ] &}, {b,
     p // Divisors // Select[#, #^2 >= p &] &}] // Reap // Last // Last
{a1, a2, c} =.;