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A224301 Sorted areas of primitive integer Heronian triangles.

%I A224301 #36 Nov 14 2019 15:46:31
%S A224301 6,12,12,24,30,36,36,42,60,60,60,60,66,72,84,84,84,84,90,90,114,120,
%T A224301 120,120,126,126,126,132,156,156,168,168,168,180,180,198,204,210,210,
%U A224301 210,210,210,210,216,234,240,252,252,252,264,264,270,288,300,300,306
%N A224301 Sorted areas of primitive integer Heronian triangles.
%C A224301 The sequence gives the sorted areas of primitive triangles which have integer side lengths and integer areas.
%H A224301 Giovanni Resta, <a href="/A224301/b224301.txt">Table of n, a(n) for n = 1..10000</a>
%H A224301 Michael Somos, <a href="http://grail.eecs.csuohio.edu/~somos/tritab.html">Heronian Triangle Table</a>
%e A224301 The smallest Heronian triangle is (3,4,5) as perimeter and area are integers. The first term of the sequence is thus the area of this triangle, which is 6.
%t A224301 AMax=400;
%t A224301 Do[
%t A224301   c=p/b;
%t A224301   a1=Sqrt[b^2+c^2+2Sqrt[b^2c^2-4A^2]];
%t A224301   a2=Sqrt[b^2+c^2-2Sqrt[b^2c^2-4A^2]];
%t A224301   If[IntegerQ[a2]&&GCD[a2,b,c]==1&&a1>a2>=b,A//Sow(*{A,a2,b,c}//Sow*)];
%t A224301   If[IntegerQ[a1]&&GCD[a1,b,c]==1,A//Sow(*{A,a1,b,c}//Sow*)];
%t A224301   ,{A,6,AMax,6}
%t A224301   ,{p,4A^2//Divisors//Select[#,EvenQ[#]&&#>=2A&]&//#/2+2A^2/#&//Select[#,IntegerQ]&}
%t A224301   ,{b,p//Divisors//Select[#,#^2>=p&]&}
%t A224301 ]//Reap//Last//Last
%t A224301 {a1,a2,c}=.;
%t A224301 (* _Albert Lau_, May 20 2016 *)
%Y A224301 Cf. A120131, A120132, A120133, A072294, A188158.
%K A224301 nonn
%O A224301 1,1
%A A224301 _Mihir Mathur_, Apr 03 2013
%E A224301 Definition corrected by _Giovanni Resta_, Apr 03 2013
%E A224301 More terms from _Giovanni Resta_, Apr 03 2013