A156687 Perimeters of Pythagorean triangles that can be constructed in exactly 5 different ways.
420, 660, 924, 1008, 1080, 1200, 1512, 1584, 1716, 1800, 1872, 1890, 2700, 3150, 3168, 3240, 3480, 3528, 3570, 3720, 3744, 4410, 4440, 4536, 4590, 4704, 4872, 4896, 4950, 5208, 5292, 5472, 5600, 5670, 6000, 6090, 6210, 6216, 6624, 6630, 6660, 6888
Offset: 1
Examples
As 924 is the third smallest integer that can occur as the perimeter of exactly 5 Pythagorean triples - specifically (42,440,442), (77,420,427), (132,385,407), (198,336,390) and (231,308,385) - then a(3)=924.
References
- Sierpinski, W.; Pythagorean Triangles, Dover Publications, Inc., Mineola, New York, 2003.
- Beiler, Albert H.; Recreations In The Theory Of Numbers, Chapter XIV, The Eternal Triangle, Dover Publications Inc., New York, 1964, pp. 104-134.
Links
- Ray Chandler, Table of n, a(n) for n = 1..10000
- Ron Knott, Right-angled Triangles and Pythagoras' Theorem
Programs
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Mathematica
SetSystemOptions["ReduceOptions"->{"DiscreteSolutionBound"->100000}];AllPerimeterTriples[n_Integer]/;n>0:=Module[{result=Reduce[Reduce[{x^2+y^2==z^2,z>y>x>0,Element[{x,y,z},Integers],x+y+z==n},{x,y,z}]]},If[result===False,{},Sort[{x,y,z}/.{ToRules[result]}]]];Select[Range[10000],Length[AllPerimeterTriples[ # ]]==5 &]
Comments