On orders in quadratic number fields with unusual sets of distances

Let $\mathcal{O}$ be an order in an algebraic number field and suppose that the set of distances $\mathrm{\Delta }(\mathcal{O})$ of $\mathcal{O}$ is nonempty (equivalently, $\mathcal{O}$ is not half-factorial). If $\mathcal{O}$ is seminormal (in particular, if $\mathcal{O}$ is a principal order), then $\min \mathrm{\Delta }(\mathcal{O})=1$. So far, only a few examples of orders were found with $\min \mathrm{\Delta }(\mathcal{O})>1$. We say that $\mathrm{\Delta }(\mathcal{O})$ is unusual if $\min \mathrm{\Delta }(\mathcal{O})>1$. In the present paper, we establish algebraic characterizations of orders $\mathcal{O}$ in real quadratic number fields with $\min \mathrm{\Delta }(\mathcal{O})>1$. We also provide a classification of the real quadratic number fields that possess an order whose set of distances is unusual. As a consequence thereof, we revisit certain squarefree integers (cf. OEIS A135735) that were studied by A. J. Stephens and H. C. Williams.