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A385819 Numbers k such that there are least five primitive Heron triangles having the same area and perimeter k.

%I A385819 #19 Jul 24 2025 05:50:54
%S A385819 2842,3542,5642,5750,6314,7238,7546,9790,15470,15778,17710,20026,
%T A385819 21658,21970,22610,26962
%N A385819 Numbers k such that there are least five primitive Heron triangles having the same area and perimeter k.
%H A385819 BBS Math Blog, <a href="https://bbs.emath.ac.cn/forum.php?mod=viewthread&amp;tid=17031">Primitive Heron triangles with equal perimeter and area (in Chinese)</a>.
%e A385819 3542 is a term because there exists 5 primitive Heron triangles: {{421,1518,1603}, {511,1375,1656}, {583,1288,1671},{759,1096,1687}, {851,1001,1690}} with same perimeter 3542 and same area 318780.
%e A385819 20026 is a term because there exists 6 primitive Heron triangles: {{2108,8493,9425}, {2173,8398,9455}, {2261,8277,9488}, {2418,8075,9533}, {4123,6205,9698}, {4588,5729,9709}} with same perimeter 20026 and same area 8410920.
%t A385819 sol = Association[];
%t A385819 For[n = 2, n <= 6000, n += 2,
%t A385819 For[z = Ceiling[n/3], z < Floor[n/2], z++,
%t A385819 For[x = 1, x < Floor[n/3], x++, y = n - x - z;
%t A385819    If[x + y > z > y > x && GCD[x, y, z] == 1, p = (x + y + z)/2;
%t A385819     A = Sqrt[p (p - x) (p - y) (p - z)];
%t A385819     If[IntegerQ[A], d = ToString@n <> "->" <> ToString@A; t = {x, y, z};
%t A385819      If[KeyExistsQ[sol, d], AppendTo[sol[d], t], sol[d] = {t}]]]]]];
%t A385819 Select[sol, Length@# > 4 &]
%Y A385819 Cf. A051516, A070138, A188158
%K A385819 nonn,hard,more
%O A385819 1,1
%A A385819 _Zhining Yang_, Jul 09 2025