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

3, 7, 10, 11, 14, 15, 17, 19, 23, 26, 30, 31, 35, 39, 42, 43, 46, 47, 51, 55, 58, 59, 62, 67, 69, 71, 74, 77, 78, 79, 82, 83, 87, 91, 94, 95, 97, 103, 105, 106, 107, 110, 111, 113, 115, 119, 122, 123, 127, 130, 131, 138, 139, 142, 143, 151, 154, 155, 158, 159, 163, 165, 167, 170

Offset: 1

Author

Frank M Jackson, Aug 08 2024

Keywords

• nonn

Comments

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.

Examples

			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.
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.
		

Links

• Frank M Jackson, Table of n, a(n) for n = 1..1787

Crossrefs

Cf. A005117, A006991.

Programs

Mathematica

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]]