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A375017 Squarefree numbers k such that k is the area of a rational isosceles triangle.

%I A375017 #32 Mar 30 2025 04:39:10
%S A375017 3,7,10,11,14,15,17,19,23,26,30,31,35,39,42,43,46,47,51,55,58,59,62,
%T A375017 67,69,71,74,77,78,79,82,83,87,91,94,95,97,103,105,106,107,110,111,
%U A375017 113,115,119,122,123,127,130,131,138,139,142,143,151,154,155,158,159,163,165,167,170
%N A375017 Squarefree numbers k such that k is the area of a rational isosceles triangle.
%C A375017 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 A375017 Frank M Jackson, <a href="/A375017/b375017.txt">Table of n, a(n) for n = 1..1787</a>
%e A375017 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 A375017 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 A375017 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 A375017 Cf. A005117, A006991.
%K A375017 nonn
%O A375017 1,1
%A A375017 _Frank M Jackson_, Aug 08 2024