Analytical expressions are derived for the number of fractions with equal numerators in the Farey sequence of order n, Fₙ, and in the truncated Farey sequence Fₙ1/k containing all Farey fractions below $1/k$, with 1≤ k ≤ n. These developments lead to an expression for the rank of $1/k$ in Fₙ, or equivalently |Fₙ1/k|, and to remarkable relations between the ranks of different unit fractions. Furthermore, the results are extended to Farey fractions of the form $2/k$.
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Rogelio Tomás (2024) studied this question.
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