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.