A280073 Numerator of Farey fractions sorted according to increasing k, with k = numerator + denominator. Fractions with same k are sorted in order of increasing denominator.

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

Offset

1,5

Comments

The parameter k is the Manhattan 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, 1/5, 3/4, 2/5, 1/6, 3/5, 1/7, 4/5, 2/7, 1/8, ..., .

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..104.

Mathematica

nmax = 25;
(* fracs are fractions represented in the triangle with vertices
(0, 1), (1, nmax) and (nmax, nmax) *)
fracs = Sort@Union@Flatten@Table[a/b, {b, nmax}, {a, 0, b}];
(* Sorting generated fractions according to increasing Manhattan distance first, and then by increasing denominator *)
fracsorted =
  SortBy[fracs, {Numerator@# + Denominator@# &, Denominator@# &}];
nmaxlimit = Floor[(1/6)* nmax^2]; (* Safe limit for a correctly sorted sequence since asymptotically half of the generated fractions can be properly sorted according to Manhattan distance *)
Take[Numerator@fracsorted, nmaxlimit]

Crossrefs

Cf. A279782, A279783.

Sequence in context: A324830 A141110 A325758 * A327523 A190770 A292149

Adjacent sequences: A280070 A280071 A280072 * A280074 A280075 A280076

Keyword

nonn,frac

Author

Robert G. Wilson v and Andres Cicuttin, Dec 25 2016

Status

approved