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Sorted areas of primitive integer Heronian triangles.
7

%I #36 Nov 14 2019 15:46:31

%S 6,12,12,24,30,36,36,42,60,60,60,60,66,72,84,84,84,84,90,90,114,120,

%T 120,120,126,126,126,132,156,156,168,168,168,180,180,198,204,210,210,

%U 210,210,210,210,216,234,240,252,252,252,264,264,270,288,300,300,306

%N Sorted areas of primitive integer Heronian triangles.

%C The sequence gives the sorted areas of primitive triangles which have integer side lengths and integer areas.

%H Giovanni Resta, <a href="/A224301/b224301.txt">Table of n, a(n) for n = 1..10000</a>

%H Michael Somos, <a href="http://grail.eecs.csuohio.edu/~somos/tritab.html">Heronian Triangle Table</a>

%e 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 AMax=400;

%t Do[

%t c=p/b;

%t a1=Sqrt[b^2+c^2+2Sqrt[b^2c^2-4A^2]];

%t a2=Sqrt[b^2+c^2-2Sqrt[b^2c^2-4A^2]];

%t If[IntegerQ[a2]&&GCD[a2,b,c]==1&&a1>a2>=b,A//Sow(*{A,a2,b,c}//Sow*)];

%t If[IntegerQ[a1]&&GCD[a1,b,c]==1,A//Sow(*{A,a1,b,c}//Sow*)];

%t ,{A,6,AMax,6}

%t ,{p,4A^2//Divisors//Select[#,EvenQ[#]&&#>=2A&]&//#/2+2A^2/#&//Select[#,IntegerQ]&}

%t ,{b,p//Divisors//Select[#,#^2>=p&]&}

%t ]//Reap//Last//Last

%t {a1,a2,c}=.;

%t (* _Albert Lau_, May 20 2016 *)

%Y Cf. A120131, A120132, A120133, A072294, A188158.

%K nonn

%O 1,1

%A _Mihir Mathur_, Apr 03 2013

%E Definition corrected by _Giovanni Resta_, Apr 03 2013

%E More terms from _Giovanni Resta_, Apr 03 2013