Let $K$ be a totally real number field and $S$ a non-trivial subsemiring of the ring of integers of $K$, and denote by $H$ the monoid of non-zero elements of $S$ under multiplication. The set of all non-zero sequences $\mathbf N^+ \to S$ can be made into a commutative, cancellative monoid $D_S$ by endowing it with the operation of Dirichlet convolution. It is clear that $D_S$ is factorial only if so is $H$. But is there any case at all in which $S$ is a ring and $D_S$ is factorial (respectively, atomic)? I don't even know the answer for the rational case.

A monoid is called atomic if every non-unit element is a (finite) product of atoms (aka irreducible elements) in at least one way, and factorial if every non-unit element can be expressed as a product of atoms in an essentially unique way.