## What are the prime divisors of $\det(A_n)$, where $A_n$ is the $n\times n$ matrix given by $(A_n)_{i,j}={n\choose|i-j|}$?

For $$n\in\mathbb{Z}_{\ge 1}$$, let $$A_n$$ be the $$n\times n$$ matrix given by $$(A_n)_{i,j}={n\choose |i-j|}$$. From this post it is clear that $$\det(A_n)=\prod_{k=0}^{n-1}\left[\left(\exp\left(\frac{2\pi k i}{n}\right)+1\right)^n-1\right]=\prod_{k=0}^{n-1}\left(2^n(-1)^k\cos^n\left(\frac{\pi k}n\right)-1\right).$$ Also, $$\det(A_n)$$ is obviously integer for all $$n$$.

Question: What is known about the (prime) divisors of $$\det(A_n)$$?

If this is too broad, I am particularly interested in pairs $$(p,d)$$ where $$p$$ is prime with $$d\mid p-1$$ and $$p\nmid\det(A_{(p-1)/d})$$.

## Asymptotic estimation of $A_n$

Let $$A_n$$ represent the number of integers that can be written as the product of two element of $$[[1,n]]$$.

I am looking for an asymptotic estimation of $$A_n$$.

First, I think it’s a good start to look at the exponent $$\alpha$$ such that :

$$A_n = o(n^\alpha)$$

I think we have : $$2 < alpha$$. To prove this lower bound we use the fact that the number of primer numbers $$\leq n$$ is about $$\frac{n}{\log n}$$. Hence we have the trivial lower bound (assuming $$n$$ is big enough) :

$$\frac{n}{\log n} \cdot \binom{ E(\frac{n}{\log n})}{2} = o(n^3)$$

Now is it possible to get a good asymptotic for $$A_n$$ and not just this lower bound ? Is what I’ve done so far correct ?

Thank you !