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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. 3
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 (list; graph; refs; listen; history; text; internal format)
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

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Last modified August 3 22:03 EDT 2021. Contains 346441 sequences. (Running on oeis4.)