Cannonball Polygons with Multiplicities

We generalize the Cannonball Problem by introducing integer-valued and non-increasing arithmetic functions $w$. We associate these functions $w$ with certain polygons, which we call cannonball polygons. Through this correspondence, we show that for any $Z\in \mathbb{N}$, there exists a cannonball polygon with multiplicity 8 and largest side of length $Z$. Moreover, for any multiplicity $s$ greater than 8, we provide an asymptotic formula for the number of distinct classes of cannonball polygons with multiplicity $s$.