Divisor sequences of atoms in Krull monoids

The divisor sequence of an irreducible element ($\mathit{\text{atom}}$) $a$ of a reduced monoid $H$ is the sequence $(s_n{)}_{n\in \mathbb{N}}$ where, for each positive integer $n$, $s_n$ denotes the number of distinct irreducible divisors of $a^n$. In this work we investigate which sequences of positive integers can be realized as divisor sequences of irreducible elements in Krull monoids. In particular, this gives a means for studying non-unique direct-sum decompositions of modules over local Noetherian rings for which the Krull-Remak-Schmidt property fails.