OFFSET
4,1
COMMENTS
We say that two fractions p/q and r/s are similarly ordered if (p-r)(q-s) >= 0. If p_1/q_1, p_2/q_2, ... is the (ordered) sequence of Farey fractions of order n, then a(n) is the largest integer m such that p_k/q_k and p_l/q_k are similarly ordered for all k and l > k with l - k <= m.
Since the Farey sequence of order n < 4 only contains fractions that are similarly ordered, a(n) does not exist if n < 4.
LINKS
Thomas Bloom, Problem 1005, Erdős Problems.
Paul Erdös, A note on Farey series, The Quarterly Journal of Mathematics 13:1 (1943), pp. 82-85.
Erdős problems database contributors, Erdős problem database, see no. 1005.
A. E. Mayer, On neighbours of higher degree in Farey series, The Quarterly Journal of Mathematics 13:1 (1942), pp. 185-192.
Wouter van Doorn, Improved bounds for the Mayer-Erdős phenomenon on similarly ordered Farey fractions, arXiv:2509.00121 [math.NT], 2025.
FORMULA
a(n) < n/4 + 5 (for all n >= 4), while a(n) >= (n/12)*(1 - 4/n^(1/3)).
Conjecture: For all n >= 92, a(n) = floor(n/4) + d, where d is either 1, 2, 2 or 4, depending on whether n is congruent to 0, 1, 2, or 3 modulo 4 [van Doorn (2025)].
EXAMPLE
The Farey sequence of order 4 is: 0/1, 1/4, 1/3, 1/2, 2/3, 3/4, 1/1. Here, 1/4 and 2/3 are the only fractions that are not similarly ordered, and there are exactly two other fractions (namely 1/3 and 1/2) in between them. That is why a(4) = 2.
CROSSREFS
KEYWORD
nonn
AUTHOR
Wouter van Doorn, Sep 03 2025
STATUS
approved