Are there any natural numbers $n$ that satisfy the condition $7921\sigma(n) = 15840n$? Are there any natural numbers $n$ that satisfy the condition $7921\sigma(n) = 15840n$, where $\sigma(n)$ denotes the sum of divisors of $n$?
This question arises from the theory of immaculate groups (or, equivalently, Leinster groups). An immaculate group is a group, such that its order is equal to the sum of all orders of its proper normal subgroups.
It is easy to see, that if $A$ is a non-abelian simple group then $A\times\mathbb{Z}_n$ is immaculate iff $(|A|+1)\sigma(n) = 2|A|n$. Two well known examples of immaculate groups of that form are $A_5\times\mathbb{Z}_{15128}$ and $A_6\times\mathbb{Z}_{366776}$. In terms of immaculate groups this question thus can be reworded as:
"Does there exist such $n$, that $M_{11}\times\mathbb{Z}_n$ is immaculate?", where $M_{11}$ stands for Mathieu simple group of order $7920$.
Currently I know only two facts about such $n$-s: if they exist, then $7921|n$, and that such $n$-s, if they exist, are too large to be found by exhaustive search.
Any help will be appreciated.
 A: This is not an answer either: I have written a program doing a non-exhaustive search, and so far, no solutions have come up for the Mathieu group $M_{11}$; however, the program did find a solution e.g. for the larger Mathieu group $M_{22}$ of order $443520=2^7\cdot3^2\cdot5\cdot7\cdot11$:
Let $n=55009909630=2\cdot5\cdot13\cdot79\cdot109\cdot157\cdot313$, then since $$
\begin{align}
(|M_{22}|+1)\cdot\sigma(n)&=443521\cdot\sigma(2\cdot5\cdot13\cdot79\cdot109\cdot157\cdot313) \\
&=13\cdot109\cdot313\cdot3\cdot6\cdot14\cdot80\cdot110\cdot158\cdot314 \\
&=2^9\cdot3^2\cdot5^2\cdot7\cdot11\cdot13\cdot79\cdot109\cdot157\cdot313 \\
&=2\cdot|M_{22}|\cdot n\text{,}\end{align}$$
$M_{22}\times\mathbb{Z}_{55009909630}$ must be immaculate.
A: Perhaps the solution to this problem might be realized via an application of the latest results of Holdener, et. al. on abundancy outlaws?
Holdener, Weiner (2014) - Searching for and Characterizing Abundancy Outlaws
Holdener, Moore (2011) - A Geometric Representation of the Abundancy Index
Holdener, Czarnecki (2008) - The Abundancy Index: Tracking Down Outlaws
Holdener, Stanton (2007) - Abundancy "Outlaws" of the Form $\frac{\sigma(N) + t}{N}$
