  The divisor sequence of an irreducible element (\textit{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.
