A375017 - OEIS

%I #29 Aug 16 2024 21:09:42

%S 3,7,10,11,14,15,17,19,23,26,30,31,35,39,42,43,46,47,51,55,58,59,62,

%T 67,69,71,74,77,78,79,82,83,87,91,94,95,97,103,105,106,107,110,111,

%U 113,115,119,122,123,127,130,131,138,139,142,143,151,154,155,158,159,163,165,167,170

%N Squarefree numbers k such that k is the area of a rational isosceles triangle.

%C All rational isosceles triangles are the join of two identical rational right triangles along one of their common legs. Therefore, for any g, a squarefree congruent number, it can be the area of the right triangle creating a rational isosceles triangle with area 2g. If g is odd then the rational isosceles triangle will have squarefree k = 2g. If g is even then the rational isosceles triangle can be reduced by a factor 4 to give a squarefree value for k = g/2.

%H Frank M Jackson, <a href="/A375017/b375017.txt">Table of n, a(n) for n = 1..1787</a>

%e The congruent number 5 can create a rational right triangle with sides (9/6, 40/6, 41/6)) and squarefree area 5. This can create a rational isosceles triangle with sides (3, 41/6, 41/6) or (80/6, 41/6, 41/6) with squarefree area 10.

%e However the congruent number 6 can create a rational right triangle with sides (3, 4, 5)) and squarefree area 6. This can create a rational isosceles triangle with sides (5/2, 5/2, 3) or (4, 5/2, 5/2) with squarefree area 3.

%t lst = Last /@ReadList["https://oeis.org/A006991/b006991.txt", {Number, Number}]; lst1={}; Do[If[EvenQ[lst[[n]]], AppendTo[lst1, lst[[n]]/2], AppendTo[lst1, 2 lst[[n]]]], {n, 1, Length@lst}]; (Sort@lst1)[[1 ;; 75]]

%Y Cf. A005117, A006991.

%K nonn

%O 1,1

%A _Frank M Jackson_, Aug 08 2024