On power values of sum of divisors function in arithmetic progressions

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