On integers of the forms \(k\pm 2^{n}\) and \(k2^{n}\pm 1\) (Q884517)

The author studies certain properties of integers of the forms \(k\pm 2^n\) and \(k2^n\pm 1\). A typical result is the following: there exists an infinite arithmetic progression of positive odd numbers such that for every term \(k\) of it and for every integer \(n\geq 0\) each of the integers \(k\pm 2^n\) and \(k2^n\pm 1\) has at least two distinct odd prime factors. Some related open problems and conjectures are also posed.