Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: A Study of @-numbers   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.

21
The categories of this paper are: adap-org
14
The categories of this paper are: math-ph
37
The categories of this paper are: supr-con
24
The categories of this paper are: dg-ga
1
The categories of this paper are: cs
30
The categories of this paper are: q-alg
11
The categories of this paper are: hep-lat
4
The categories of this paper are: cond-mat
9
The categories of this paper are: nlin
5
The categories of this paper are: gr-qc
6
The categories of this paper are: astro-ph
8
The categories of this paper are: hep-ex
3
The categories of this paper are: physics
29
The categories of this paper are: funct-an
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The categories of this paper are: alg-geom
32
The categories of this paper are: atom-ph