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

1,1

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.

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.

Formula

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

Example

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.

Mathematica

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

Prog

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

Crossrefs

Supersequence of A027862.

Sequence in context: A228922 A025490 A216824 * A087938 A103729 A234739

Adjacent sequences: A230809 A230810 A230811 * A230813 A230814 A230815

Keyword

nonn

Author

Frank M Jackson, Oct 30 2013

Status

approved