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.

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, 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, 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

Offset: 1

Author

Andres Cicuttin, May 26 2023

Keywords

• nonn

Examples

			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.
		

Links

• Eric Weisstein's World of Mathematics, Farey Sequence.
• Wikipedia, Farey Sequence.

Crossrefs

Cf. A005728, A262669, A262670, A363321.

Programs

Mathematica

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