  We study the simple Bershadsky-Polyakov algebra $\mathcal W_k =
\mathcal{W}_k(sl_3,f_{\theta})$ at positive integer levels and classify their
irreducible modules. In this way we confirm the conjecture from
arXiv:1910.13781. Next, we study the case $k=1$. We discover that this vertex
algebra has a Kazama-Suzuki-type dual isomorphic to the simple afine vertex
superalgebra $L_{k'} (osp(1 \vert 2))$ for $k'=-5/4$. Using the free-field
realization of $L_{k'} (osp(1 \vert 2))$ from arXiv:1711.11342, we get a
free-field realization of $\mathcal W_k$ and their highest weight modules. In a
sequel, we plan to study fusion rules for $\mathcal W_k$.
