Let $\gcd(i,j)$ and $lcm(i,j)$ be the greatest common divisor and least common multiple of the pair of positive integers $i$ and $j$. Denote the sum of divisors of $k$ by $\sigma(k)=\sum_{d\vert k}d$.

I noticed that the following is known: if $M_n$ is the $n\times n$ matrix with entries $M_n(i,j)=\sigma(\gcd(i,j))$ then $\det(M_n)=n!$.

By analogy, define the matrix $A_n$ with entries $A_n(i,j)=\sigma(lcm(i,j))$. I could not find anything about this, so I ask:

QUESTION. Is there an explicit evaluation of the determinant $\det(A_n)$?

Examples. For $n\geq1$, here are few terms of $\det(A_n)$: $$1, -6, 72, -672, 20160, 1451520, -81285120, 1393459200.$$ These numbers do have small primes factors, so it seems reasonable to expect a closed form.

$\newcommand{\lcm}{\operatorname{lcm}}$Let $\gcd(i,j)$ and $\lcm(i,j)$ be the greatest common divisor and least common multiple of the pair of positive integers $i$ and $j$. Denote the sum of divisors of $k$ by $\sigma(k)=\sum_{d\vert k}d$.

I noticed that the following is known: if $M_n$ is the $n\times n$ matrix with entries $M_n(i,j)=\sigma(\gcd(i,j))$ then $\det(M_n)=n!$.

By analogy, define the matrix $A_n$ with entries $A_n(i,j)=\sigma(\lcm(i,j))$. I could not find anything about this, so I ask:

QUESTION. Is there an explicit evaluation of the determinant $\det(A_n)$?

Examples. For $n\geq1$, here are few terms of $\det(A_n)$: $$1, -6, 72, -672, 20160, 1451520, -81285120, 1393459200.$$ These numbers do have small primes factors, so it seems reasonable to expect a closed form.