Farey sequence

plural Farey sequences

1. mathematicsFor a given positive integer n, the sequence of completely reduced fractions between 0 and 1 which, when in lowest terms, have denominators less than or equal to n, arranged in order of increasing size.

   Riemann sums have also important connections with various problems from number theory, among them the Riemann Hypothesis, through their link with Farey sequences.

   2009, Michel Weber, Dynamical Systems and Processes, European Mathematical Society, page 549

   2 more examples

   Students should then see the number of fractions N, in the Farey sequence is equal to #92;phi(2)#43;#92;phi(3)#43;#92;phi(4)#43;#92;dots#43;#92;phi(n), where #92;phi(n) is the number of positive integers less than or equal to n that are relatively prime to n.

   2002, Alfred S. Posamentier, Jay Stepelman, Teaching Secondary Mathematics, Merrill, page 403

   2007, Jakub Pawlewicz, Order Statistics in the Farey Sequences in Sublinear Time, Lars Arge, Michael Hoffmann, Emo Welzl (editors), Algorithms - ESA 2007: 15th Annual European Symposium, Proceedings, Springer, LNCS 4698, page 218, The Farey sequence of order n (denoted ℱₙ) is the increasing sequence of all irreducible fractions from interval [0,1] with denominators less than or equal to n. The Farey sequences have numerous interesting properties and they are well known in the number theory and in the combinatorics.