A280315 Denominator of Farey fractions sorted according to increasing k, with k = numerator + denominator. Fractions with same k are sorted in order of increasing denominator.

1, 1, 2, 3, 3, 4, 5, 4, 5, 6, 5, 7, 5, 7, 8, 7, 9, 6, 7, 8, 9, 10, 7, 11, 7, 8, 9, 10, 11, 12, 9, 11, 13, 8, 11, 13, 14, 9, 11, 13, 15, 9, 10, 11, 12, 13, 14, 15, 16, 11, 13, 17, 10, 11, 12, 13, 14, 15, 16, 17, 18, 11, 13, 17, 19, 11, 13, 16, 17, 19, 20, 13, 15, 17, 19, 21, 12, 13, 14, 15, 16, 17

Offset

1,3

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..82.

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[Denominator@fracsorted, nmaxlimit]

Crossrefs

Cf. A280073, A279782, A279783.

Sequence in context: A071585 A328801 A106500 * A120245 A120246 A361261

Adjacent sequences: A280312 A280313 A280314 * A280316 A280317 A280318

Keyword

nonn,frac

Author

Robert G. Wilson v and Andres Cicuttin, Dec 31 2016

Status

approved