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.
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
KEYWORD
nonn
AUTHOR
Vladimir Joseph Stephan Orlovsky, Jul 07 2009
EXTENSIONS
Definition clarified by R. J. Mathar, Aug 14 2009
STATUS
approved