On Power Values of Sum of Divisors function in Arithmetic Progressions

TL;DR

This paper investigates the conditions under which the sum of divisors of numbers in an arithmetic progression can be perfect powers, extending previous results to more general sequences and establishing infinite occurrence under certain conditions.

Contribution

It generalizes known results about perfect power values of sum of divisors to numbers in arithmetic progressions with specific coprimality conditions.

Findings

Existence of infinitely many numbers in certain arithmetic progressions with sum of divisors as perfect powers.

Conditions relating to coprimality and modular order for the results to hold.

Either no perfect power values or infinitely many such values occur in the progression.

Abstract

Let $a\geq 1, b\geq 0$ and $k\geq 2$ be any given integers. It has been proven that there exist infinitely many natural numbers $m$ such that sum of divisors of $m$ is a perfect $k$th power. We try to generalize this result when the values of $m$ belong to any given infinite arithmetic progression $an+b$. We prove if $a$ is relatively prime to $b$ and order of $b$ modulo $a$ is relatively prime to $k$ then there exist infinitely many natural numbers $n$ such that sum of divisors of $an+b$ is a perfect $k$th power. We also prove that, in general, either sum of divisors of $an+b$ is not a perfect $k$th power for any natural number $n$ or sum of divisors of $an+b$ is a perfect $k$th power for infinitely many natural numbers $n$.