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 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 {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.