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
Keywords
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 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.
Links
- Eric Weisstein's World of Mathematics, Farey Sequence.
- Wikipedia, Farey Sequence.
Programs
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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}]
Formula
Conjecture: lim_{n->infinity} a(n)/A005728(n) = 0.
Comments