What is known about the average growth rate of the denominators of $n$ Egyptian fractions summing to one?

Motivation

In the following question posted here on MO and over at MSE, user Noah Schweber asks about a weighted count on Egyptian fraction representations (EFRs). To that end, he defines the following set of EFRs for a rational number $$q$$:

$$\mathsf{E}(q)=\left\{X\in[\mathbb{N}]^{\mathit{fin}}: q=\sum_{x\in X}{1\over x}\right\}.$$ Subsequently, he defines the function $$\mathscr{E}\colon q\mapsto \sum_{X\in \mathsf{E}(q)} \left( \prod_{x\in X}{1\over x}\right).$$

Let's call $$\mathscr{E}(1)$$ the Egyptian Schweber constant. Noah himself points out that a lower bound for this constant can be obtained quite easily. For instance, we have $$\mathscr{E}(1) > \frac{37}{36}$$, as $$\mathsf{E}(1)$$ contains both $$\{1\}$$ and $$\{2,3,6 \}$$.

I am interested in obtaining upper bounds for the Egyptian Schweber constant. Below, I'll outline what I've tried to obtain such a bound.

Aproach

My idea of solving this problem is to first use a bound on the amount of possible equation Egyptian fractions of length $$n$$. To that end, it might be somewhat easier to slightly tweak Schweber's definition to take into account the length of the EFRs: $$\mathsf{E}_{n}(q)=\left\{X_n := \{x_1 , x_2 , \dots , x_n \}: \sum_{k=1}^{n} \frac{1}{x_{k}} = q, \ 0

Then, according to p.8 of the following paper by Ronald Graham we have $$| \mathsf{E}_{n}(1) | < c_{0}^{(1+\epsilon)2^{n-1} },$$ where $$c_{0} \approx 1.264.$$

Next, we need to obtain a good lower bound for the average size of the denominators of $$\mathsf{E}_{n}(1)$$. By definition, it amounts to the sum of the denominators, divided by their amount. Let's call the average size of the denominators $$AV(n)$$, and the sum $$S(n)$$. We then have $$AV(n) = S(n)/n.$$

Using the aforementioned source, we also find that $$AV(n) > c_{0}^{-(1+\epsilon)2^{n-1} } \sum_{X_{n} \in \mathsf{E}_{n}(1)} \frac{1}{n} \sum_{x \in X_{n}} x .$$

Moreover, we then have the following upper bound for the Egyptian Schweber constant: $$\mathscr{E}(1) < \sum_{n=1}^{\infty} c_{0}^{(1+\epsilon)2^{n-1}} \times \left( \frac{1}{AV(n)} \right)^{n} .$$

Unfortunately, I haven't been able to find any sources on the average size of the denominators of the EFRs. I did find that Sylvester's sequence can give Egyptian fractions of length $$n$$ with the largest denominators. Also, I found the following relevant master's thesis, but I don't think any bounds for the average size of the EFRs are given in there either.

Question

What is known about the asymptotics (especially lower bounds) of the average size - or the sum - of the denominators of Egyptian fraction representations of one?

Note: I've also asked this question on MSE

• Bounds for individual denominators are discussed at mathoverflow.net/q/233336 Commented Oct 14, 2023 at 13:29
• @MaxAlekseyev Thank you for the link. I am not entirely sure though how to apply your results to this problem -- they give an upper bound, right? Not a lower a bound? Commented Oct 14, 2023 at 14:14
• Yes, there are only upper bounds as lower bounds for individual terms are not that interesting. E.g., in the case of $E_n(1)$ we have $x_i \geq i+1$ for $i<n$ and $x_n\geq n$. Commented Oct 14, 2023 at 15:49