Divisibility and sequence properties of σ⁺ and φ⁺

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 S₂(n), Jordan totient function Jₖ, 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 {σ⁺(n)}ₙ₌₁^∞ and {φ⁺(n)}ₙ₌₁^∞, in particular, we prove that each of these sequences contains infinitely many arithmetic progressions of length 3.