If $q^k n^2$ is an odd perfect number with Euler prime $q$, are the following statements known to hold in general?

Question

(Note: This question has been cross-posted from MO.)

Let $\sigma$ be the classical sum-of-divisors function.

A number is said to be perfect if $\sigma(N)=2N$.

If $q^k n^2$ is an odd perfect number with Euler prime $q$ (i.e., $q \equiv k \equiv 1 \pmod 4$ and $\gcd(q,n)=1$), are the following statements known to hold in general?

(a) $\gcd(n^2, \sigma(n^2))$ is large.

(b) The deficiency $D(n^2) = 2n^2 - \sigma(n^2)$ is large.

(c) The index $i(q^k) = \sigma(N/q^k)/q^k$ is large.

Using the trivial relationships:

$$\gcd(n^2, \sigma(n^2)) = \frac{D(n^2)}{\sigma(q^{k-1})} = i(q^k) = \frac{2n^2}{\sigma(q^k)}$$

then if the Descartes-Frenicle-Sorli conjecture that $k = 1$ is true, it is possible to show that a lower bound for the quantities (a), (b) and (c) is given by $n/\sqrt{3}$. (Here, I have used Acquaah and Konyagin's estimate $q < n\sqrt{3}$. The inequality $q^k < n^2$ then gives the desired large numerical bound if we use known lower bounds for the odd perfect number $N = q^k n^2$, latest of which are by Ochem and Rao.)

What happens when $k > 1$? I do know that $$\frac{\sigma(N/q^k)}{q^k} = \sigma(n^2)/q^k \geq 315$$ by using a result of Broughan, Delbourgo, and Zhou.

Is it possible to do better than this, apart from attempting a proof of (obviously) $q^k < n$?

Answer 1

Let $N = q^k n^2$ be an odd perfect number with Euler prime $q$.

In a recent preprint, Brown claims a complete proof for $q < n$, and a partial proof that the inequality $q^k < n$ holds under many cases. (See arXiv.)

In particular, since $q \leq q^k < n$ holds if Brown's proofs are correct (and completed), then the resulting lower bound is $$\gcd(n^2, \sigma(n^2)) = \frac{D(n^2)}{\sigma(q^{k-1})} = i(q^k) = \frac{2n^2}{\sigma(q^k)} > \frac{8}{5}\cdot\frac{n^2}{q^k} > \frac{8}{5}\cdot{n}.$$

Notice that Brown's proofs hold unconditionally (i.e., even if $k=1$).

We then have the desired large numerical lower bound $$\gcd(n^2, \sigma(n^2)) = \frac{D(n^2)}{\sigma(q^{k-1})} = \frac{\sigma(N/q^k)}{q^k} > \frac{8}{5}\cdot{n} > \frac{8}{5}\cdot{{10}^{500}},$$ which is an easy consequence of Ochem and Rao's $N > {10}^{1500}$ and the inequality $q^k < n$.