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

%I A162597 #6 Dec 10 2016 22:35:10
%S A162597 5,13,17,25,29,37,41,53,61,65,73,85,89,97,101,109,113,137,145,149,157,
%T A162597 173,181,185,193,197,221,229,233,241,257,265,269,277,281,293,313,317,
%U A162597 325,337,349,353,365,373,377,389,397,401,409,421,433,445,449,457,461
%N A162597 Ordered hypotenuses of primitive Pythagorean triangles, A008846, which are not hypotenuses of non-primitive Pythagorean triangles with any shorter legs.
%C A162597 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.
%C A162597 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.
%e A162597 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.
%e A162597 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.
%t A162597 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
%Y A162597 Cf. A004613, A008846.
%K A162597 nonn
%O A162597 1,1
%A A162597 _Vladimir Joseph Stephan Orlovsky_, Jul 07 2009
%E A162597 Definition clarified by _R. J. Mathar_, Aug 14 2009