On primitive covering numbers

In 2007, Zhi-Wei Sun defined a covering number to be a positive integer L such that there exists a covering system of the integers where the moduli are distinct divisors of L greater than 1. A covering number L is called primitive if no proper divisor of L is a covering number. Sun constructed an infinite set L of primitive covering numbers, and he conjectured that every primitive covering number must satisfy a certain condition. In this paper, for a given L is an element of L, we derive a formula that gives the exact number of coverings that have L as the least common multiple of the set M of moduli, under certain restrictions on M. Additionally, we disprove Sun's conjecture by constructing an infinite set of primitive covering numbers that do not satisfy his primitive covering number condition.