

A230812


Smallest squarefree side lengths of primitive integer Soddyian triangles.


3



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
(list;
graph;
refs;
listen;
history;
text;
internal format)



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(sc) = 1/sqrt(sb)+1/sqrt(sa) 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), 16.


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



