A230812 Smallest squarefree side lengths of primitive integer Soddyian triangles.

5, 13, 41, 61, 85, 113, 145, 181, 221, 265, 313, 365, 421, 481, 545, 613, 685, 761, 1013, 1105, 1201, 1301, 1405, 1513, 1741, 1861, 1985, 2113, 2245, 2381, 2521, 2665, 2813, 2965, 3121, 3281, 3445, 3613, 3785, 3961, 4141, 4513, 4705, 5101, 5305, 5513, 5941

Offset: 1

Author

Frank M Jackson, Oct 30 2013

Keywords

• nonn

Comments

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.

Examples

			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.
		

Links

• Charles R Greathouse IV, Table of n, a(n) for n = 1..10000
• F. M. Jackson, Soddyian triangles, Forum Geom. 13 (2013), 1-6.

Crossrefs

Supersequence of A027862.

Programs

Mathematica

lst = {}; Do[If[SquareFreeQ[(m+1)^2+m^2], AppendTo[lst, (m+1)^2+m^2]], {m, 1, 100}]; lst

PARI

select(issquarefree, vector(1000,m,(m+1)^2+m^2)) \\ Charles R Greathouse IV, Oct 31 2013

Formula

Squarefree integers of the form (m+1)^2+m^2 for any integer m > 0.