 Solution

Use the hook criterion in a beta set: for a finite beta set of a partition $X$, each hook length is a difference $b-c$ with $b\in X$, $0\le c<b$ and $c\notin X$, and every such difference is a hook length.

A hook length $pr$ therefore gives a bead $b$ and a gap $c=b-pr$. Along the progressionthe first position is a bead and the last is a gap. Some adjacent positions are consequently a bead followed by a gap. Their difference is $p$, so $p$ is a hook length. Similarly the progression from $b$ to $c$ in steps of $r$ gives a hook length $r$. Every position used is nonnegative because it lies between $c$ and $b$.

Thus the divisor closure of hook lengths yieldsNeither $p$ nor $r$ needs to be a prime number.