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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.

%I A343769 #8 Oct 09 2021 07:41:54
%S A343769 12,126,624,1260,1800,2100,2850,4536,5292,5580,8820,9900,12600,12642,
%T A343769 14850,15600,17640,19110,21756,23400,24948,25200,25536,28350,47040,
%U A343769 47304
%N 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.
%H A343769 Jason Zimba, <a href="https://jzimba.blogspot.com/2021/04/there-are-infinitely-many-rectangular.html">There are infinitely many rectangular Heronian triangles</a>.
%e A343769 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.
%e A343769 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.
%t A343769 (* Adapted from Albert Lau's program for A224301 *)
%t A343769 AMax = 10000;
%t A343769 Do[c = p/b;
%t A343769     a1 = Sqrt[b^2 + c^2 + 2 Sqrt[b^2 c^2 - 4 A^2]];
%t A343769     a2 = Sqrt[b^2 + c^2 - 2 Sqrt[b^2 c^2 - 4 A^2]];
%t A343769     If[IntegerQ[a2] && GCD[a2, b, c] == 1 &&
%t A343769       a1 > a2 >= b && (per = a2 + b + c;
%t A343769        IntegerQ[(per + Sqrt[per^2 - 16 A])/4]), A // Sow(*{A,a2,b,c}//
%t A343769      Sow*)];
%t A343769     If[IntegerQ[a1] &&
%t A343769       GCD[a1, b, c] == 1 && (per = a1 + b + c;
%t A343769        IntegerQ[(per + Sqrt[per^2 - 16 A])/4]), A // Sow(*{A,a1,b,c}//
%t A343769      Sow*)];, {A, 6, AMax, 6}, {p,
%t A343769      4 A^2 // Divisors //
%t A343769         Select[#, EvenQ[#] && # >= 2 A &] & // #/2 + 2 A^2/# & //
%t A343769       Select[#, IntegerQ] &}, {b,
%t A343769      p // Divisors // Select[#, #^2 >= p &] &}] // Reap // Last // Last
%t A343769 {a1, a2, c} =.;
%Y A343769 Subsequence of A224301.
%K A343769 nonn,more
%O A343769 1,1
%A A343769 _Jason Zimba_, Apr 28 2021