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A156687 Perimeters of Pythagorean triangles that can be constructed in exactly 5 different ways. 0
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 (list; graph; refs; listen; history; text; internal format)
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

Table of n, a(n) for n=1..42.

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. A099831, A099832, A099833, A009129, A010814

Sequence in context: A200521 A189982 A070237 * A191934 A147775 A169825

Adjacent sequences:  A156684 A156685 A156686 * A156688 A156689 A156690

KEYWORD

easy,nice,nonn

AUTHOR

Ant King, Feb 18 2009

STATUS

approved

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Last modified June 24 08:33 EDT 2017. Contains 288697 sequences.