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A230812 Smallest squarefree side lengths of primitive integer Soddyian triangles.

%I A230812 #15 Oct 31 2013 20:33:10
%S A230812 5,13,41,61,85,113,145,181,221,265,313,365,421,481,545,613,685,761,
%T A230812 1013,1105,1201,1301,1405,1513,1741,1861,1985,2113,2245,2381,2521,
%U A230812 2665,2813,2965,3121,3281,3445,3613,3785,3961,4141,4513,4705,5101,5305,5513,5941
%N A230812 Smallest squarefree side lengths of primitive integer Soddyian triangles.
%C A230812 A Soddyian triangle is a triangle whose outer Soddy circle has degenerated into a straight line. Its side lengths are related by the equation 1/sqrt(s-c) = 1/sqrt(s-b)+1/sqrt(s-a) where the sides a <= b <= c and s is the semiperimeter. It is Heronian. The smallest side length of a primitive Soddyian triangle is given as a = n^2((m+n)^2+m^2) for integers m >= n > 0 with GCD(m, n) = 1. If this side length is squarefree, then n = 1 and (m+1)^2+m^2 has to be squarefree. a(n) is the ordered sequence of squarefree integers t of the form t = (m+1)^2+m^2. Note that t uniquely determines the primitive Soddyian triple whenever the smallest side length is squarefree.
%H A230812 Charles R Greathouse IV, <a href="/A230812/b230812.txt">Table of n, a(n) for n = 1..10000</a>
%H A230812 F. M. Jackson, <a href="http://forumgeom.fau.edu/FG2013volume13/FG2013index.html">Soddyian triangles</a>, Forum Geom. 13 (2013), 1-6.
%F A230812 Squarefree integers of the form (m+1)^2+m^2 for any integer m > 0.
%e A230812 a(3)=41 because the triangle with sides (41, 416, 425) is a primitive Soddyian triangles, 41 is squarefree and is the 3rd occurrence of such a squarefree integer.
%t A230812 lst = {}; Do[If[SquareFreeQ[(m+1)^2+m^2], AppendTo[lst, (m+1)^2+m^2]], {m, 1, 100}]; lst
%o A230812 (PARI) select(issquarefree, vector(1000,m,(m+1)^2+m^2)) \\ _Charles R Greathouse IV_, Oct 31 2013
%Y A230812 Supersequence of A027862.
%K A230812 nonn
%O A230812 1,1
%A A230812 _Frank M Jackson_, Oct 30 2013