cp's OEIS Frontend

This is a front-end for the Online Encyclopedia of Integer Sequences, made by Christian Perfect. The idea is to provide OEIS entries in non-ancient HTML, and then to think about how they're presented visually. The source code is on GitHub.

Showing 1-3 of 3 results.

A262670 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 the average.

Original entry on oeis.org

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, 0, 2, 0, 2, 0, 2, 0, 0, 0, 0, 2, 2, 2, 0, 0, 0, 0, 0, 0, 0, 2, 2, 0, 2, 2, 2, 0, 2, 2, 2, 0, 0, 0, 0, 4, 0, 0, 0, 6, 0, 0, 2, 4, 2, 0, 2, 0, 0, 2, 2, 0, 0, 2, 0, 0, 0, 0, 2, 0, 0, 2, 0, 2, 0, 2, 2, 2, 2, 0, 2, 2, 0, 0, 0, 0, 0, 0, 0, 0
Offset: 0

Views

Author

Robert G. Wilson v, Nov 09 2015

Keywords

Comments

Because the Farey fractions are symmetrical about 1/2 for n > 1, a(n) is always even.
First occurrence of k by index, or -1 if no such occurrence exists: 0, 1, 2, -1, 60, -1, 64, -1, 207, -1, 1047, -1, 1084, -1, ..., .
Where 0 occurs: 0, 3, 7, 8, 10, 12, 13, 14, 17, 20, 22, 23, 26, 28, 30, 32, 33, ..., ;
Where 2 occurs: 2, 4, 5, 6, 9, 11, 15, 16, 18, 19, 21, 24, 25, 27, 29, 31, 36, 37, 38, ..., ;
Where 4 occurs: 60, 68, 120, 129, 148, 158, 159, 168, 180, 216, 225, 231, 239, 241, 249, ..., ;
Where 6 occurs: 65, 227, 401, 403, 492, 600, 616, 780, 861, 862, 865, 967, 1019, 1054, ..., ;
Where 8 occurs: 208, 1210, 1367, 1803, 1804, 1841, 1866, 2397, 2864, 3281, 3443, 3724, ..., ;
Where 10 occurs: 1048, 1094, 1632, 1949, 2269, 2571, 2710, 3365, 3555, 3558, 3613, 3939, ..., ;
Where 12 occurs: 1085, 1358, 2541, 3251, 4411, ..., ;
Where 18 occurs: 4830, ..., ;
For the first 5001 terms: 3315 zeros, 1 one, 1138 twos, 414 fours, 96 sixes, 19 eights, 12 tens, 5 twelves and 1 eighteen.

Examples

			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.

References

  • 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.

Links

Crossrefs

Cf. A002088, A005728, A006843, A262669.

Programs

  • Mathematica
    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, 65]

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

Views

Author

Andres Cicuttin, May 26 2023

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 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

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}]

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

Views

Author

Andres Cicuttin, May 27 2023

Keywords

Comments

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.

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

Crossrefs

Cf. A005728, A262669, A262670, A363300.

Programs

  • 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.
Showing 1-3 of 3 results.

Started in 1964 by Neil J. A. Sloane | Maintained by The OEIS Foundation Inc.

Content is available under The OEIS End-User License Agreement.