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

TL;DR

This paper explores divisibility properties and sequence behaviors of modified divisor and totient functions, extending classical conjectures and proving the existence of infinitely many arithmetic progressions within these sequences.

Contribution

It introduces new divisibility problems related to $\sigma^+$ and $\varphi^+$ functions and proves the existence of infinitely many arithmetic progressions of length three in their sequences.

Findings

Sequences $\sigma^+(n)$ and $\varphi^+(n)$ contain infinitely many arithmetic progressions of length 3.

Extended classical conjectures to new functions like $\sigma^+$ and $\varphi^+$.

Established properties of these sequences related to divisibility and progression patterns.

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 $3$.