An arithmetic conjecture on an arctangent sum

Abstract

A sequence \(x_n\), defined in terms of a sum of arctangent values, satisfies the nonlinear recurrence \(x_n = (n + x_{n-1})/(1 - nx_{n-1})\), with \(x_1 = 1\), which has been conjectured not to be an integer for \(n \geq 5\). This problem is analyzed here in terms of divisibility questions of an associated sequence. Properties of this new sequence are employed to prove that the subsequences \(\{x_{19n+5} : n \in \mathbb{N}\}\) and \(\{x_{19n+13} : n \in \mathbb{N}\}\) contain no integer values.