A Study of @-numbers

Abstract

This paper deals more generally with @-numbers defined as follows: Call `\textit{alpha number}' of order $(\underline{\alpha},\bar{\alpha})\in\mathbb{H}^2$, (denote its family by @$_{(\underline{\alpha},\bar{\alpha})\in\mathbb{H}^2; \mathcal{A}\subset \mathbb{N}}$) any $n\in\mathcal{A}\subset \mathbb{N}$ satisfying $\sigma_{\underline{\alpha}}(n) = \alpha n^{\bar{\alpha}}$ where $\sigma_{\underline{\alpha}}(n)$ is sum of divisors function and $\alpha\in\mathbb{H}$, the set of \textit{quaternions}. Specifically, if integer $n$ is such that $\alpha=\alpha_1/\alpha_2,\ \alpha_1,\alpha_2\in\mathbb{Z}^+$ with $1\leq\max(\alpha_1, \alpha_2) \le \omega(n),$ $\ \le \tau (n), \ < n$ (where $\omega(n)$ is the number of distinct prime factors of $n$, $\tau (n)$ is the number of factors of $n$), then $n$ is respectively called strong, weak or very weak alpha number. We give some examples and conjecture that there is no odd strong alpha number of order $(1,1)$. The truthfulness of this assertion implies that there is no odd perfect and certain odd multi-perfect numbers. We give all the strong even alpha numbers of order $(1,1)$ below $10^5$ and then show that there is no odd strong alpha number of order $(1,1)$ below $10^5$, using some of our results motivated by some results of Ore and Garcia. With computer search this bound can easily be surpassed. In this paper, using Rossen, Schonfield and Sandor's inequalities, in addition to the aforementioned definition, we also bound the quotient $\alpha_1/\alpha_2 =\alpha$ of order $(1,1)$, though a very weak bound. Some areas for future research are also pointed out as recommendations.