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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.

%I A363321 #23 Feb 16 2025 08:34:05
%S A363321 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,
%T A363321 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,
%U A363321 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
%N 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.
%C A363321 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.
%H A363321 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/FareySequence.html">Farey Sequence</a>.
%H A363321 Wikipedia, <a href="https://en.wikipedia.org/wiki/Farey_sequence">Farey Sequence</a>.
%F A363321 Conjecture: lim_{n->infinity} a(n)/A005728(n) = 0.
%e A363321 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.
%t A363321 a[n_]:= Module[{len, fn, sn},
%t A363321 fn = FareySequence[n];
%t A363321 len = Length[fn];
%t A363321 sn = Range[0, len - 1]/(len - 1);
%t A363321 Count[fn - sn, 0]];
%t A363321 Table[a[j], {j, 1, 80}]
%Y A363321 Cf. A005728, A262669, A262670, A363300.
%K A363321 nonn
%O A363321 1,1
%A A363321 _Andres Cicuttin_, May 27 2023