Arithmetical completeness for some extensions of the pure logic of necessitation

We investigate the arithmetical completeness theorems of some extensions of Fitting, Marek, and Truszczy\'{n}ski's pure logic of necessitation $\mathbf{N}$. For $m,n \in \omega$, let $\mathbf{NA}_{m,n}$, which was introduced by Kurahashi and Sato, be the logic obtained from $\mathbf{N}$ by adding the axiom scheme $\Box^n A \to \Box^m A$. In this paper, among other things, we prove that for each $m,n \geq 1$, the logic $\mathbf{NA}_{m,n}$ becomes a provability logic, that is, there exists a provability predicate $\mathrm{Pr}_T(x)$ of $T$ whose $T$-verifiable modal principles are exactly the logic $\mathbf{NA}_{m,n}$.