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A096467 Numbers that can be the longest side of a primitive Heronian triangle.

%I A096467 #25 Feb 16 2025 08:32:53
%S A096467 5,6,8,13,15,17,20,21,24,25,26,28,29,30,35,36,37,39,40,41,42,44,45,48,
%T A096467 50,51,52,53,55,56,58,60,61,63,65,66,68,69,70,73,74,75,77,80,82,85,87,
%U A096467 88,89,90,91,92,93,95,96,97,100,101,102,104,105,106,109,110,111,112,113
%N A096467 Numbers that can be the longest side of a primitive Heronian triangle.
%C A096467 Here a primitive Heronian triangle has integer sides a,b,c with gcd(a,b,c) = 1 and integral area. Note that all primes of the form 4k+1 are in this sequence. It appears that a prime of the form 4k+3 is never the longest side of a Heronian triangle. Cheney's article contains many theorems about these triangles.
%H A096467 Ray Chandler, <a href="/A096467/b096467.txt">Table of n, a(n) for n = 1..10000</a> (first 240 terms from Vincenzo Librandi)
%H A096467 Wm. Fitch Cheney, Jr., <a href="http://www.jstor.org/stable/2300173">Heronian Triangles</a>, Amer. Math. Monthly, Vol. 36, No. 1 (Jan 1929), 22-28.
%H A096467 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/HeronianTriangle.html">Heronian Triangle</a>
%e A096467 5 is on this list because the triangle with sides 3, 4, 5 has integral area.
%t A096467 nn=150; lst={}; Do[s=(a+b+c)/2; If[IntegerQ[s] && GCD[a, b, c]==1, area2=s(s-a)(s-b)(s-c); If[area2>0 && IntegerQ[Sqrt[area2]], AppendTo[lst, a]]], {a, nn}, {b, a}, {c, b}]; Union[lst]
%Y A096467 Cf. A083875 (area/6 of primitive Heronian triangles), A096468 (perimeter of primitive Heronian triangles).
%Y A096467 Cf. A239246, A306626.
%K A096467 nonn
%O A096467 1,1
%A A096467 _T. D. Noe_, Jun 22 2004