

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
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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



