A279782 Numerator of Farey fractions sorted according to increasing k, with k = numerator^2 + denominator^2. Fractions with same k are sorted in order of increasing denominator.

0, 1, 1, 1, 2, 1, 3, 1, 2, 3, 1, 4, 1, 2, 3, 5, 4, 1, 3, 5, 1, 6, 2, 5, 4, 1, 5, 3, 7, 1, 2, 7, 3, 4, 8, 1, 5, 7, 6, 5, 7, 1, 2, 3, 9, 8, 4, 7, 5, 1, 9, 6, 3, 7, 10, 5, 1, 2, 8, 4, 9, 1, 11, 3, 10, 7, 9, 5, 8, 11, 1, 2, 3, 7, 4, 12, 5, 11, 6, 1, 9, 7, 11, 5, 8

Offset

1,5

Comments

The parameter k is the square of the Euclidean distance of the corresponding points to the origin in the "denominator, numerator" representation space.

The fractions in order begin: 0/1, 1/1, 1/2, 1/3, 2/3, 1/4, 3/4, 1/5, 2/5, 3/5, 1/6, 4/5, 1/7, 2/7, 3/7, 5/6, ..., .

Note that the fraction 2/4 is not in the above since it can be reduced to 1/2.

Links

Table of n, a(n) for n=1..85.

Example

The point (1,0) lies closer to the origin than any other point in the first quadrant. Therefore a(1) = 0;
The next point is (1,1) because it lies closer to the origin. Therefore a(2) = 1;
The point (2,1) lies the next closest to the origin. Therefore a(3) = 1;
The point (3,1) is the next closest to the origin. Therefore a(4)= 1;
The point (3,2) is the next closest to the origin. Therefore a(5)= 2;
etc.
The fractions 4/7 and 1/8 are respectively represented by the points (7,4) and (8,1), both points are at the same distance sqrt(65) from the origin, but 4/7 has smaller denominator than 1/8 hence a(17)=4 and a(18)=1.

Mathematica

nmax = 22; (* Maximum explorative denominator *)
fracs = Sort@Union@Flatten@Table[a/b, {b, nmax}, {a, 0, b}];
(* Sorting generated fractions according to increasing Euclidean distance first, and then by increasing denominator *)
fracsorted = SortBy[fracs, {(Numerator@#)^2 + (Denominator@#)^2 &,
    Denominator@# &}];
nmaxlimit = Floor[nmax^2/4] (*Safe limit for a correctly sorted sequence*);
Take[Numerator@fracsorted, nmaxlimit]

Crossrefs

Cf. A279783.

Sequence in context: A255809 A332604 A132662 * A358921 A132589 A054843

Adjacent sequences: A279779 A279780 A279781 * A279783 A279784 A279785

Keyword

nonn,frac

Author

Robert G. Wilson v and Andres Cicuttin, Dec 18 2016

Status

approved