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A375017
Squarefree numbers k such that k is the area of a rational isosceles triangle.
2
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
1,1
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.
LINKS
EXAMPLE
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.
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]]
CROSSREFS
Sequence in context: A095947 A335488 A285036 * A345168 A348612 A349799
KEYWORD
nonn
AUTHOR
Frank M Jackson, Aug 08 2024
STATUS
approved