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A224301
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Sorted areas of primitive integer Heronian triangles.
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7
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6, 12, 12, 24, 30, 36, 36, 42, 60, 60, 60, 60, 66, 72, 84, 84, 84, 84, 90, 90, 114, 120, 120, 120, 126, 126, 126, 132, 156, 156, 168, 168, 168, 180, 180, 198, 204, 210, 210, 210, 210, 210, 210, 216, 234, 240, 252, 252, 252, 264, 264, 270, 288, 300, 300, 306
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OFFSET
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1,1
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COMMENTS
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The sequence gives the sorted areas of primitive triangles which have integer side lengths and integer areas.
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LINKS
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EXAMPLE
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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.
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MATHEMATICA
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AMax=400;
Do[
c=p/b;
a1=Sqrt[b^2+c^2+2Sqrt[b^2c^2-4A^2]];
a2=Sqrt[b^2+c^2-2Sqrt[b^2c^2-4A^2]];
If[IntegerQ[a2]&&GCD[a2, b, c]==1&&a1>a2>=b, A//Sow(*{A, a2, b, c}//Sow*)];
If[IntegerQ[a1]&&GCD[a1, b, c]==1, A//Sow(*{A, a1, b, c}//Sow*)];
, {A, 6, AMax, 6}
, {p, 4A^2//Divisors//Select[#, EvenQ[#]&&#>=2A&]&//#/2+2A^2/#&//Select[#, IntegerQ]&}
, {b, p//Divisors//Select[#, #^2>=p&]&}
]//Reap//Last//Last
{a1, a2, c}=.;
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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EXTENSIONS
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STATUS
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approved
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