A279783 Denominator 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.

1, 1, 2, 3, 3, 4, 4, 5, 5, 5, 6, 5, 7, 7, 7, 6, 7, 8, 8, 7, 9, 7, 9, 8, 9, 10, 9, 10, 8, 11, 11, 9, 11, 11, 9, 12, 11, 10, 11, 12, 11, 13, 13, 13, 10, 11, 13, 12, 13, 14, 11, 13, 14, 13, 11, 14, 15, 15, 13, 15, 13, 16, 12, 16, 13, 15, 14, 16, 15, 13, 17, 17, 17, 16, 17, 13, 17, 14, 17, 18, 16

Offset

1,3

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..81.

Example

The point (1,0) lies closer to the origin than any other point in the first quadrant. Therefore a(1) = 1;
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) = 2;
The point (3,1) is the next closest to the origin. Therefore a(4)= 3;
The point (3,2) is the next closest to the origin. Therefore a(5)= 3;
etc.

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[Denominator@fracsorted, nmaxlimit]

Crossrefs

Cf. A279782.

Sequence in context: A350254 A196241 A086592 * A132663 A306631 A023964

Adjacent sequences: A279780 A279781 A279782 * A279784 A279785 A279786

Keyword

nonn,frac

Author

Robert G. Wilson v and Andres Cicuttin, Dec 19 2016

Status

approved