Formulas and algorithms for the length of a Farey sequence

Abstract

Abstract This paper proves several novel formulas for the length of a Farey sequence of order n. The formulas use different trade-offs between iteration and recurrence and they range from simple to more complex. The paper also describes several iterative algorithms for computing the length of a Farey sequence based on these formulas. The algorithms are presented from the slowest to the fastest in order to explain the improvements in computational techniques from one version to another. The last algorithm in this progression runs in $$O(n^{2/3})$$ O ( n 2 / 3 ) time and uses only $$O(\sqrt{n})$$ O ( n ) memory, which makes it the most efficient algorithm for computing $$|F_n|$$ | F n | described to date. With this algorithm we were able to compute the length of the Farey sequence of order $$10^{18}$$ 10 18 .