A262670 -id:A262670 - cp's OEIS frontend

A262669 Consider the Farey sequence of order n, F_n, and that the average distance between any two adjacent pairs in F_n is 1/A002088(n). Then a(n) is the number of adjacent pairs whose difference is less than the average.

0, 0, 0, 2, 2, 2, 4, 6, 8, 8, 12, 14, 18, 18, 20, 26, 28, 32, 32, 40, 42, 46, 48, 58, 58, 66, 76, 78, 84, 88, 94, 100, 106, 114, 120, 126, 128, 142, 150, 162, 166, 178, 178, 194, 200, 206, 214, 230, 236, 246, 250, 266, 274, 292, 296, 312, 322, 338, 344, 360, 360, 388, 400, 408, 416, 436

Offset: 0

Robert G. Wilson v, Sep 26 2015

nonn

Because the Farey fractions are symmetrical about 1/2, a(n) is always even.
Conjecture: this is a monotonic sequence. For n = 0, 1, 3, 4, 8, 12, 17, 23, 41 & 59, a(n) = a(n+1).
If instead the question is when the difference is equal to the average, then the sequence becomes 0, 1, 2, 0, 2, 2, 2, 0, 0, 2, 0, 2, 0, 0, 0, 2, 2, 0, 2, 2, 0, 2, 0, 0, 2, 2, ..., . A262670.
Conjecture: Twice the number of pairs less than the average (2*A262669) plus the number of pairs which equal the average (A262670) never exceed the number of pairs which are greater than the average for n greater than 245.
   f( 1000) =   100972,
   f( 2000) =   403750,
   f( 3000) =   908068,
   f( 4000) =  1614072,
   f( 5000) =  2522376,
   f( 6000) =  3631762,
   f( 7000) =  4943332,
   f( 8000) =  6456904,
   f( 9000) =  8171296,
   f(10000) = 10088132.

			a(5) = 2. F_5 = {0, 1/5, 1/4, 1/3, 2/5, 1/2, 3/5, 2/3, 3/4, 4/5, 1} and the first forward difference is {1/5, 1/20, 1/12, 1/15, 1/10, 1/10, 1/15, 1/12, 1/20, 1/5}. The average distance is 1/10 since A002088(5) = 10 which is also the number of adjacent pairs, a/b & c/d.

Albert H. Beiler, Recreations in the Theory of Numbers, The Queen of Mathematics Entertains, Chapter XVI, "Farey Tails", Dover Books, NY, 1966, pgs 168-172.

Robert G. Wilson v, Table of n, a(n) for n = 0..5000
Cut the Knot, Farey Series.
The University of Surrey, Math Dept., Fractions in the Farey Series and the Stern-Brocot Tree.
Eric Weisstein's World of Mathematics, Farey Sequence.
Wikipedia, Farey Sequence.

Cf. A002088, A005728, A006843, A262670.

f[n_] := Block[{diff = Differences@ Union@ Flatten@ Table[a/b, {b, n}, {a, 0, b}], ave = 1/Sum[ EulerPhi[ m], {m, n}]}, {Length@ Select[diff, ave < # &], Length@ Select[diff, ave == # &], Length@ Select[diff, ave > # &]}]; Array[ f[#][[1]] &, 65, 0]

a(n) = (n/Pi)^2 + O(n/3*(log(n))^(2/3)*(log(log(n)))^(4/3)), (A. Walfisz 1963).

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.

Original entry on oeis.org

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

Andres Cicuttin, May 26 2023

nonn

			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.

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

Cf. A005728, A262669, A262670, A363321.

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}]

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.

Original entry on oeis.org

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

Andres Cicuttin, May 27 2023

nonn

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.

			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.

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

Cf. A005728, A262669, A262670, A363300.

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}]

Conjecture: lim_{n->infinity} a(n)/A005728(n) = 0.

cp's OEIS Frontend

A262669 Consider the Farey sequence of order n, F_n, and that the average distance between any two adjacent pairs in F_n is 1/A002088(n). Then a(n) is the number of adjacent pairs whose difference is less than the average.

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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.

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Mathematica

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.

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Author

Keywords

Comments

Examples

Links

Crossrefs

Programs

Mathematica

Formula