Divisibility and Sequence Properties of $\sigma^+$ and $\varphi^+$

Abstract

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