The sequence \(\{x_n\}\) defined by $$x_n = \frac{n+x_{n-1}}{1-nx_{n-1}},$$
with \(x_1=1\), appeared in the context of some arctangent sums. We
establish the fact that \(x_n\neq 0\) for \(n \geq 4\) and conjecture that \(x_n\) is not an integer for \(n \geq 5\). This conjecture is given a combinatorial
interpretation in terms of Stirling numbers via the elementary symmetric
functions. The problem features linkage with a well-known conjecture on
the existence of infinitely many primes \(1+n^2\), as well as our conjecture
that \((1 + 1^2)(1 + 2^2)\cdots (1 + n^2)\) is not a square for \(n > 3\). We present
an algorithm that verifies the latter for \(n = 10^{3200}\).
