On power values of sum of divisors function in arithmetic progressions

Abstract

Let $a\ge 1, b\ge 0$ and $k\ge 2$ 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 kth 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 kth power. We also prove that, in general, either sum of divisors of $an+b$ is not a perfect kth power for any natural number n or sum of divisors of $an+b$ is a perfect kth power for infinitely many natural numbers n.