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

1,1

Comments

For any given N we can always find at least N Pythagorean triangles with the same perimeter.

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

Example

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.

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 &]

Crossrefs

Cf. A098714, A099831, A099832, A099833, A009129, A010814.

Sequence in context: A189982 A070237 A305416 * A191934 A147775 A169825

Adjacent sequences: A156684 A156685 A156686 * A156688 A156689 A156690

Keyword

easy,nice,nonn

Author

Ant King, Feb 18 2009

Status

approved