A162597 Ordered hypotenuses of primitive Pythagorean triangles, A008846, which are not hypotenuses of non-primitive Pythagorean triangles with any shorter legs.

5, 13, 17, 25, 29, 37, 41, 53, 61, 65, 73, 85, 89, 97, 101, 109, 113, 137, 145, 149, 157, 173, 181, 185, 193, 197, 221, 229, 233, 241, 257, 265, 269, 277, 281, 293, 313, 317, 325, 337, 349, 353, 365, 373, 377, 389, 397, 401, 409, 421, 433, 445, 449, 457, 461

Offset

1,1

Comments

Hypotenuses of primitive Pythagorean triangles are shown in A008846 and A020882, and may also be hypotenuses of non-primitive Pythagorean triangles (see A009177, A118882). The sequence contains those hypotenuses of A008846 where in the set of Pythagorean triangles with this hypotenuse the one with the shortest leg is a primitive one.

This ordering first on hypotenuses, then filtering on the shortest legs, and then selecting the primitive triangles removes 125, 169, 205, 289, 305, 425, etc. from A008846.

Links

Table of n, a(n) for n=1..55.

Example

The hypotenuse 25 appears in the triangle 25^2 = 7^2 + 24^2 (primitive) and in the triangle 25^2 = 15^2 + 20^2 (non-primitive). The triangle with the shortest leg (here: 7) is primitive, so 25 is in the sequence.
The hypotenuse 125 appears in the triangles 125^2 = 35^2 + 120^2 (non-primitive), 125^2 = 44^2 + 117^2 (primitive), 125^2 = 75^2 + 100^2 (non-primitive). The case with the shortest leg (here: 35) of these 3 is not primitive, so 125 is not in the sequence.

Mathematica

f[n_]:=Module[{k=1}, While[(n-k^2)^(1/2)!=IntegerPart[(n-k^2)^(1/2)], k++; If[2*k^2>=n, k=0; Break[]]]; k]; lst1={}; Do[If[f[n^2]>0, a=f[n^2]; b=(n^2-a^2)^(1/ 2); If[GCD[n, a, b]==1, AppendTo[lst1, n]]], {n, 3, 6!}]; lst1

Crossrefs

Cf. A004613, A008846.

Sequence in context: A081804 A004613 A008846 * A120960 A198440 A094194

Adjacent sequences: A162594 A162595 A162596 * A162598 A162599 A162600

Keyword

nonn

Author

Vladimir Joseph Stephan Orlovsky, Jul 07 2009

Extensions

Definition clarified by R. J. Mathar, Aug 14 2009

Status

approved