 Farey sequence

For a positive integer $Q$, the Farey sequence ${\mathcal{F}}_Q$ is the increasing list of reduced fractions in $[0,1]$ with denominators at most $Q$, including $0/1$ and $1/1$. For example,Each denominator $q\ge 2$ contributes exactly $\phi (q)$ new interior fractions, because the allowable numerators are precisely those coprime to $q$. Therefore the number of terms is $1+{\sum }_{q=1}^Q\phi (q)=1+\Phi (Q)$, using the summatory totient function.

For two distinct fractions $a/b<c/d$ in the list, their difference is $(bc-ad)/(bd)\ge 1/Q^2$, since the integer $bc-ad$ is positive. This arithmetic separation, rather than arbitrary real spacing, makes Farey fractions useful in rational approximation. The Stern-Brocot tree organizes the same positive reduced fractions by mediant insertion instead of a denominator cutoff.

A Farey sequence, denoted as \( F_n \), is a sequence of completely reduced fractions between 0 and 1 that have denominators less than or equal to a given positive integer \( n \). The Farey sequence is arranged in increasing order. Each fraction in the sequence is expressed in simplest form, meaning that the numerator and denominator are coprime (they have no common factors other than 1).