Dividing and using some identities of [1] I've deduced the following facts, see also the remarks below. After these introductory paragraphs, to motivate our question, I am asking if we can deduce some useful information for odd perfect numbers $n$ from the literature evaluating convolutions sums.
Fact. A) If $n$ is an odd perfect number of the form $36m+9$ then is satisfied the following identity $$4\sigma\left(\frac{2n}{3}\right)=3\sigma\left(\frac{2n}{9}\right)+2(n+\sigma(n)),\tag{1}$$ and additionally also the congruence $\sigma_3(n)\equiv 18\text{ mod }36$ holds. B) If our odd perfect number $n$ has the form $12m+1$ then satisfies $$3\sigma\left(\frac{n-1}{12}\right)+\sigma\left(\frac{3(n-1)}{4}\right)=4\sigma\left(\frac{\sigma(n)-2}{8}\right),\tag{2}$$ and also the congruences $\sigma_3(n)\equiv 6\text{ mod }12$ and $\sigma_5(n)\equiv 2\text{ mod }12$ hold.
Remarks about previous facts. My belief is that the opposite direction of $(1)$, and the other equation $(2)$ are true, but I cann't get these to prove that each equation is a characterization of an odd perfect number of the given form. I tried work invoking Euler's theorem for odd perfect numbers $n=2^{\alpha}3^{\beta}m$ such that $(2,m)=(3,m)=1$ by cases, to deduce from the statement $(1)$ a contradiction, but if my calculations are rigths it was failed: the case 1 ($\beta=2$ and $\alpha=0$) yields $\sigma(m)=\frac{18}{13}m$; the case 2 ($\beta>2$ and $\alpha=0$) will be $(3^{\beta +1}-1)\sigma(m)=4\cdot 3^\beta m$; the case 3 ($\beta=2$ and $\alpha\geq 1$) was $\sigma(m)=\frac{2^{\alpha+1}\cdot 9m}{13}$ and finally the case 4 ($\beta>2$ and $\alpha\geq 1$) yields $\sigma(m)=\frac{2^{\alpha+2}\cdot 3^{\beta}}{3^{\beta+1}-1}$. And I cann't find counterexamples, I don't find integers $n\equiv 0\text{ mod }9$ satisfying $(1)$ and I don't find any integer $n\equiv 1\text{ mod }12$ and such that $8\mid (-2+\sigma(n))$, satisfying $(2)$. (If my congruences are rights) I have not used much specific information about odd perfect numbers to deduce these congruences.
Question. Is it possible to deduce statements for odd perfect numbers, under the assumption that these exist, using convolution sums involving divisor functions $\sigma_k(n)$ or other arithmetic functions? I am asking about congruences, analysis of some convolution sums for the case of a specialization related to odd perfect numbers, I say useful identities, or congruences or specializations with applications in the study of odd perfect numbers. Many thanks.
If you want follow up the approach in my fact for your answer it is welcome: try to calculate special equations for odd perfect numbers or to study similar congruences for different divisor functions $$\sigma_{k}(n)\equiv a\text{ mod }b.$$ It was my attempt/approach to collect information, about odd perfect numbers, extracted from convolution sums involving arithmetic functions.
I add here the references that I've used for to deduce my facts, I've used a specialization of an identity that is showed in the second paragraph of page 255 of [1], I add also the reference for the theorem due to Touchard's.
References:
[1] James G. Huard, Zhiming M. Ou, Blair K. Spearman, and Kenneth S. Williams, Elementary Evaluation of Certain Convolution Sums Involving Divisor Functions, Number Theory for the Millenium II, A K Peters (2002).
[2] J. Touchard, On prime numbers and perfect numbers, Scripta Math. vol 19, pp. 35-39 (1953).
