Covering systems with odd moduli

The concept of a covering system was first introduced by Erd?s in 1950. Since their introduction, a lot of the research regarding covering systems has focused on the existence of covering systems with certain restrictions on the moduli. Arguably, the most famous open question regarding covering systems is the odd covering problem. In this paper, we explore a variation of the odd covering problem, allowing a single odd prime to appear as a modulus in the covering more than once, while all other moduli are distinct, odd, and greater than 1. We also consider this variation while further requiring the moduli of the covering system to be square-free.