On the coprimeness relation from the viewpoint of monadic second-order logic

Let $\mathfrak{C}$ denote the structure of the natural numbers with the coprimeness relation. We prove that for each non-zero natural number $n$, if a $\Pi^1_n$-set of natural numbers is closed under automorphisms of $\mathfrak{C}$, then it is definable in $\mathfrak{C}$ by a monadic $\Pi^1_n$-formula of the signature of $\mathfrak{C}$ having exactly $n$ set quantifiers. On the other hand, we observe that even a much weaker version of this property fails for certain expansions of $\mathfrak{C}$.