A New Classification of Positive Integers Via New Divisor Functions

Let d₁ = 1 < d₂ < d₃ < ⋯ < dτ(n) = n denote the increasing sequence of the divisors of a positive integer n. In this paper, for real or complex values of $α$, we define and study some properties of two new divisor functions σe,α and σo,α. The first computes the sum of the $α$-th powers of the divisors of n with even indices, and the second computes the sum of the $α$-th powers of the divisors of n with odd indices. We also introduce a new type of positive integers, namely, k-index ratio numbers and state three conjectures related to them.