Arithmetical properties of a sequence arising from an arctangent sum Original Research Article

The sequence {xn} defined by xn=(n+xn−1)/(1−nxn−1), with x1=1, appeared in the context of some arctangent sums. We establish the fact that xn≠0 for ngreater-or-equal, slanted4 and conjecture that xn is not an integer for ngreater-or-equal, slanted5. 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 of the form n2+1, as well as our conjecture that (1+12)(1+22)cdots, three dots, centered(1+n2) is not a square for n>3. We present an algorithm that verifies the latter for nless-than-or-equals, slant103200.