FareySequence

FareySequence[n]

generates the Farey sequence of order n.

FareySequence[n,k]

gives the k^(th) element of the Farey sequence of order n.

Details

  • The Farey sequence of order n is the sorted sequence of completely reduced fractions between 0 and 1 with denominators not exceeding n.

Examples

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Basic Examples  (1)

Generate the Farey sequence of order 5:

Find the 17^(th) element of the Farey sequence of order 24:

Applications  (4)

Farey arc diagram, connecting adjacent rationals in a Farey sequence:

Show them together:

Visualize a pattern of denominators of a Farey sequence of order 12:

The length of a Farey sequence of few small orders:

Compare to a closed-form formula in terms of Euler's totient function:

Construct Ford circles from a Farey sequence:

Properties & Relations  (2)

Obtain a Farey sequence as a union of Subdivide lists:

FareySequence[n,k] is equivalent to FareySequence[n]k:

FareySequence[n,k] is much faster:

Introduced in 2014
 (10.0)
 |
Updated in 2016
 (10.4)