Arithmetical properties of a sequence arising from an arctangent sum
The authors study the sequence \(x_n=(n+x_{n-1})/(1-nx_{n-1})\) with \(x_1=1\), which appeared in the context of arctangent sums. The authors prove that \(x_n \neq 0\) for \(n \geq 4\), and conjecture that \(x_n\) is not an integer for \(n \geq 5\). The authors also conjecture that \(\prod_{i=1}^n (i^2+1)\) is not a square, for \(n \geq 3\) , and give partial results. This has meanwhile been proved by \textit{J. Cilleruelo} [J. Number Theory 128, No. 8, 2488--2491 (2008; Zbl 1213.11057)].
scientific article; zbMATH DE number 5285748