Speaker
Description
Throughout, we let $d_1 = 1 < d_2 < d_3 < \cdots < d_{\tau(n)} = n$ denote the increasing sequence of the divisors of a positive integer $n$. We define the arithmetic functions $\sigma_{e}$ and $\sigma_{o}$ by:
\begin{equation}
\sigma_{e}(n) = \sum_{i=1, \, 2 \mid i}^{\tau(n)} d_i \text{ and }\sigma_{o}(n) = \sum_{i=1, \, 2 \nmid i}^{\tau(n)} d_i.
\end{equation}
Note that the sequences $\sigma_{e}(n)\ (n\in\mathbb{N})$ and $\sigma_{o}(n)\ (n\in\mathbb{N})$ correspond to the sequences A086989 and A086988 in the On-Line Encyclopedia of Integer Sequences, respectively. A positive integer $n$ such that $\sigma_{o}(n)$ divides $\sigma(n)$ (equivalently $\sigma_{o}(n)$ divides $\sigma_{e}(n)$ since $\sigma(n)=\sigma_{o}(n)+\sigma_{e}(n)$) is called an index ratio number; these integers form sequence A392467 in the OEIS.
The following definition introduces a new classification of positive integers based on the functions $\sigma_{e}$ and $\sigma_{o}$.
*For any integer $n\ge2$, define the arithmetic function
$$
\kappa(n):=\dfrac{\sigma_{e}(n)}{\sigma_{o}(n)}\in\mathbb{Q}_{>0}.
$$
For a positive rational number $k$, we say that $n$ is a $k$-\emph{index ratio number} if $\kappa(n)=k$, and we set
$$*
G_k:=\{\,n\in\mathbb{N}\setminus\{1\} : \kappa(n)=k\,\}.
$$
For $n=1$ we have $\sigma_e(1)=0$ and $\sigma_o(1)=1$, so the quotient would be $0$ rather than a positive rational number (and $d_2$ does not exist). Thus every integer $n\ge2$ belongs to exactly one set $G_k$, namely $G_{\kappa(n)}$, and the nonempty sets among the $G_k$ ($k\in\mathbb{Q}_{>0}$) form a partition of $\mathbb{N}\setminus{1}$. Hence,
$$
A392467=\{1\}\cup\bigcup_{k\in\mathbb{N}}G_k .
$$
Our talk is divided into three parts. In the first, we study the $k$-index ratio numbers $n$ in the case where $\tau(n)$ is even; the remaining case, in which $\tau(n)$ is odd, is examined in the second part. The third part collects several questions arising from these results as open problems for future research.