About Farey Sequences

For any positive integer \(n\), the Farey sequence \(F_n\) is the set of irreducible rational numbers \(\dfrac{a}{b}\) with \(0\leq a\leq b\leq n\) and \(\mathrm{gcd}(a,b)=1\) arranged in increasing order. The first three Farey sequences are: \[ \begin{array}{lcl} F_1 & = & \displaystyle{\left\{\frac{0}{1},\frac{1}{1}\right \}} \\ F_2 & = & \displaystyle{\left\{\frac{0}{1},\frac{1}{2},\frac{1}{1}\right \}} \\ F_3 & = & \displaystyle{\left\{\frac{0}{1},\frac{1}{3},\frac{1}{2},\frac{2}{3},\frac{1}{1}\right \}} \end{array} \]

Compute the maximum value of \(n\) for which the number of terms of \(F_n\) does not exceed \(500\).

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