A009129 -id:A009129 - cp's OEIS frontend

A098714 Only one Pythagorean triangle of this perimeter exists.

12, 24, 30, 36, 40, 48, 56, 70, 72, 80, 96, 108, 112, 126, 140, 150, 154, 156, 160, 176, 182, 192, 198, 200, 204, 208, 216, 220, 224, 228, 234, 260, 276, 286, 306, 308, 320, 324, 340, 348, 350, 352, 364, 372, 374, 378, 380, 384, 392, 400, 416, 418, 442, 444

Offset: 1

Marcus Rezende (marcus(AT)anp.gov.br), Sep 29 2004

nonn

Previous name was : This is the perimeter (n) of square triangles with integer sides and that have only a single solution.

Numbers in A010814 not in A009129. - Hugo Pfoertner, Mar 29 2018

Ray Chandler, Table of n, a(n) for n = 1..10000 (first 5000 terms from Hugo Pfoertner)
Project Euler, Problem 75: Singular integer right triangles.
Index entries related to Pythagorean triples.

Cf. A009129, A010814.

forstep(p=12,444,2,d=0;for(k=1,p-3,for(j=k+1,p-k-1,if(j*j+k*k==(p-j-k)^2,d++)));if(d==1,print1(p,", "))) \\ Hugo Pfoertner, Mar 29 2018

n = a + b + c; c^2=a^2+b^2; a, b, c (sides) and n (perimeter) are integers; for a given "n" there is only a single triple of a, b and c.

More terms from Hugo Pfoertner and Ray Chandler, Oct 27 2004

New name from Hugo Pfoertner, Mar 29 2018

A156687 Perimeters of Pythagorean triangles that can be constructed in exactly 5 different ways.

Original entry on oeis.org

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

Ant King, Feb 18 2009

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

			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.

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.

Ray Chandler, Table of n, a(n) for n = 1..10000
Ron Knott, Right-angled Triangles and Pythagoras' Theorem

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

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 &]

cp's OEIS Frontend

A098714 Only one Pythagorean triangle of this perimeter exists.

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