Divisibility and Sequence Properties of ${\sigma }^{+}$ and ${\phi }^{+}$

Inspired by Lehmer’s and Deaconescu’s conjectures, as well as various analogue problems concerning Euler’s totient function $\phi (n)$, Schemmel’s totient function $S_2(n)$, Jordan totient function $J_k$, and the unitary totient function ${\phi }^{\ast }(n)$, we investigate analogous divisibility problems involving the functions $\sigma (n)$, ${\sigma }^{+}(n)$, and ${\phi }^{+}(n)$. Further, we establish some interesting properties of the sequences ${\{{\sigma }^{+}(n)\}}_{n=1}^{\mathrm{\infty }}$ and ${\{{\phi }^{+}(n)\}}_{n=1}^{\mathrm{\infty }}$, in particular, we prove that each of these sequences contains infinitely many arithmetic progressions of length $3$.