A New Classification of Positive Integers Via New Divisor Functions

Let $d_1 = 1 < d_2 < d_3 < \cdots < d_{\tau(n)} = n$ denote the increasing sequence of the divisors of a positive integer $n$. In this paper, for real or complex values of $\alpha$, we define and study some properties of two new divisor functions $\sigma_{e,\alpha}$ and $\sigma_{o,\alpha}$. The first computes the sum of the $\alpha$-th powers of the divisors of $n$ with even indices, and the second computes the sum of the $\alpha$-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.