Farey fractions with equal numerators and the rank of unit fractions

Analytical expressions are derived for the number of fractions with equal numerators in the Farey sequence of order $n$, $F_n$, and in the truncated Farey sequence $F_n^{1\mathrm{/}k}$ containing all Farey fractions below $1\mathrm{/}k$, with $1\le k\le n$. These developments lead to an expression for the rank of $1\mathrm{/}k$ in $F_n$, or equivalently $\mid F_n^{1\mathrm{/}k}\mid$, and to remarkable relations between the ranks of different unit fractions. Furthermore, the results are extended to Farey fractions of the form $2\mathrm{/}k$.