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A363300 Number of fractions of the Farey sequence of order n, F_n, that can be expressed as x/y, where y = #{F_n} - 1.

%I A363300 #24 Feb 16 2025 08:34:05
%S A363300 2,3,3,5,7,9,7,3,11,9,13,3,3,9,23,25,25,23,33,17,33,23,5,49,45,5,33,
%T A363300 23,53,3,49,43,9,69,49,77,75,63,7,47,11,3,9,5,5,55,53,9,55,61,57,11,
%U A363300 97,133,67,5,81,5,7,95,15,9,5,217,13,17,75,107,133,19,113,5,21,85,117,5,9,121,3,3
%N A363300 Number of fractions of the Farey sequence of order n, F_n, that can be expressed as x/y, where y = #{F_n} - 1.
%H A363300 Eric Weisstein's World of Mathematics, <a href="https://mathworld.wolfram.com/FareySequence.html">Farey Sequence</a>.
%H A363300 Wikipedia, <a href="https://en.wikipedia.org/wiki/Farey_sequence">Farey Sequence</a>.
%e A363300 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 7 fractions (0, 1/5, 2/5, 1/2, 3/5, 4/5, 1) common to both sequences, F_5 and S_5, then a(5) = 7.
%t A363300 a[n_]:= Module[{len, fn, sn},
%t A363300 fn = FareySequence[n];
%t A363300 len = Length[fn];
%t A363300 sn = Range[0, len - 1]/(len - 1);
%t A363300 Intersection[fn, sn] // Length];
%t A363300 Table[a[j], {j,1,80}]
%Y A363300 Cf. A005728, A262669, A262670, A363321.
%K A363300 nonn
%O A363300 1,1
%A A363300 _Andres Cicuttin_, May 26 2023