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

0,4

Comments

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.

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

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.

Formula

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

Example

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.

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[#][[1]] &, 65, 0]

Crossrefs

Cf. A002088, A005728, A006843, A262670.

Sequence in context: A090858 A187504 A036654 * A345256 A291299 A241576

Adjacent sequences: A262666 A262667 A262668 * A262670 A262671 A262672

Keyword

nonn

Author

Robert G. Wilson v, Sep 26 2015

Status

approved