Divisibility and Sequence Properties of $σ^+$ and $φ^+$

Abstract

Inspired by Lehmer's and Deaconescu's conjectures, as well as various analogue problems concerning Euler's totient function $φ(n)$, Schemmel's totient function $S_{2}(n)$, Jordan totient function $J_k$, and the unitary totient function $φ^{*}(n)$, we investigate analogous divisibility problems involving the functions $σ(n)$, $σ^{+}(n)$, and $φ^{+}(n)$. Further, we establish some interesting properties of the sequences $\left\{σ^+(n)\right\}_{n=1}^\infty$ and $\left\{φ^+(n)\right\}_{n=1}^\infty$, in particular, we prove that each of these sequences contains infinitely many arithmetic progressions of length $3$.