Does $n$ divide $2^{n-1} - 1$ for an infinite number of composite $n$?

Yes, $p$ always divides $2^{p-1}-1$ for any prime $p ge 3$. Are there any odd composite $n$ for which $n$ divides $2^{n-1}-1$?

And on a related note, $p$ never divides $2^{p-1}+1$ for any prime $p ge 3$. Are there any odd composite $n$ for which $n$ divides $2^{n-1}+1$?

I ask these questions because they are natural spin-offs from some well-known questions. It is well-known that no integer $n>2$ divides $2^n-1$. And it is basic to establish $p$ always divides $2^{p-1}-1$ for any prime $p ge 3$. But for general $n$, the relationship between the integers $varphi(n)$ and $n-1$ is not clear-cut.

Thanks for considering!