A363321 Number of fractions of the Farey sequence of order n, F_n, that coincide with those of the sequence of the #{F_n} equally distributed fractions between 0 and 1.

2, 3, 3, 5, 5, 7, 5, 3, 9, 7, 9, 3, 3, 7, 17, 17, 15, 11, 13, 11, 19, 15, 5, 25, 21, 5, 11, 17, 25, 3, 7, 13, 5, 29, 27, 41, 35, 33, 7, 17, 7, 3, 5, 3, 3, 23, 17, 5, 19, 15, 25, 9, 35, 47, 29, 5, 31, 3, 7, 27, 9, 5, 5, 61, 5, 9, 23, 41, 51, 15, 29, 3, 9, 23, 31, 3, 7, 33, 3, 3

Offset

1,1

Comments

The conjectured formula below suggests that as the value of n increases, the proportion of terms in the Farey sequence F_n that align with the #F_n rationals, evenly distributed between 0 and 1, tends to decrease.

Links

Table of n, a(n) for n=1..80.

Eric Weisstein's World of Mathematics, Farey Sequence.

Wikipedia, Farey Sequence.

Formula

Conjecture: lim_{n->infinity} a(n)/A005728(n) = 0.

Example

For n = 5, we have the Farey sequence F_5 = {0, 1/5, 1/4, 1/3, 2/5, 1/2, 3/5, 2/3, 3/4, 4/5, 1} with 11 terms, and the corresponding sequence S_5 = {0, 1/10, 1/5, 3/10, 2/5, 1/2, 3/5, 7/10, 4/5, 9/10, 1} consisting of the 11 equidistant fractions {x/10} with 0 <= x <= 10. Since there are 5 fractions (0, 2/5, 1/2, 3/5, 1) in the same positions in both sequences, F_5 and S_5, then a(5) = 5.

Mathematica

a[n_]:= Module[{len, fn, sn},
fn = FareySequence[n];
len = Length[fn];
sn = Range[0, len - 1]/(len - 1);
Count[fn - sn, 0]];
Table[a[j], {j, 1, 80}]

Crossrefs

Cf. A005728, A262669, A262670, A363300.

Sequence in context: A167411 A340195 A340192 * A112823 A247176 A325163

Adjacent sequences: A363318 A363319 A363320 * A363322 A363323 A363324

Keyword

nonn,changed

Author

Andres Cicuttin, May 27 2023

Status

approved