Arithmetical properties of a sequence arising from an arctangent sum

Abstract

The sequence { x n } defined by x n = ( n + x n − 1 ) / ( 1 − n x n − 1 ) , with x 1 = 1 , appeared in the context of some arctangent sums. We establish the fact that x n ≠ 0 for n ⩾ 4 and conjecture that x n is not an integer for n ⩾ 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 of the form n 2 + 1 , as well as our conjecture that ( 1 + 1 2 ) ( 1 + 2 2 ) ⋯ ( 1 + n 2 ) is not a square for n > 3 . We present an algorithm that verifies the latter for n ⩽ 10 3200 .