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A224302 Sorted perimeters of primitive Heronian triangles.

%I A224302 #54 Feb 29 2020 02:05:15
%S A224302 12,16,18,30,32,36,36,40,42,42,44,48,50,50,54,54,54,56,60,64,64,64,66,
%T A224302 68,70,70,72,76,78,80,80,84,84,84,84,84,90,90,90,96,98,98,98,98,98,98,
%U A224302 100,100,100,104,104,108,108,108,108,108,110,112,112,112,112
%N A224302 Sorted perimeters of primitive Heronian triangles.
%C A224302 Here a primitive Heronian triangle has integer sides a,b,c with gcd(a,b,c) = 1 and integral area.
%C A224302 The perimeters of primitive Heronian triangles are even [Wenzel Šimerka, 1869]. - _Mo Li_, Feb 02 2020
%D A224302 L. E. Dickson, History of the Theory of Numbers, vol. II: Diophantine Analysis, Dover, 2005, p. 196. [21a]
%H A224302 Giovanni Resta, <a href="/A224302/b224302.txt">Table of n, a(n) for n = 1..10002</a>
%H A224302 L. E. Dickson, <a href="https://archive.org/details/HistoryOfTheTheoryOfNumbersVolII/page/n221/mode/2up">History of the Theory of Numbers, vol. II</a>, 1952, see p. 196 [21 a].
%H A224302 Michael Somos, <a href="http://grail.eecs.csuohio.edu/tritab.html">Heronian Triangle Table</a>
%H A224302 Wikipedia, <a href="http://en.wikipedia.org/wiki/Heronian_triangle">Heronian triangle</a>
%e A224302 a(1) = 12 as it is the perimeter of the Heronian triangle having sides 3,4,5 and is the smallest Heronian triangle perimeter.
%e A224302 a(2) = 16 as it is the perimeter of the Heronian triangle having sides 5,5,6 and is the 2nd smallest Heronian triangle perimeter.
%t A224302 hQ[a_, b_, c_] := IntegerQ@Sqrt@Block[{s = (a + b + c)/2}, s (s - a) (s - b) (s - c)];
%t A224302 Sort[Reap[Do[If[GCD[a, b, c] == 1 && hQ[a, b, c], Sow@(a + b + c)], {a, 100}, {b, a}, {c, a - b + 1, b}]][[2, 1]]] (* The last numbers given may not be exactly in the right place. *) (* _Jinyuan Wang_, Feb 02 2020 *)
%Y A224302 Cf. A120131, A120132, A120133, A224301.
%K A224302 nonn
%O A224302 1,1
%A A224302 _Mihir Mathur_, Apr 04 2013
%E A224302 Corrected and extended by _Giovanni Resta_, Apr 04 2013