New primitive covering numbers and their properties

A covering number is a positive integer L   such that a covering system of the integers can be constructed with distinct moduli that are divisors d>1d>1 of L. If no proper divisor of L is a covering number, then L is called primitive. In 2007, Zhi-Wei Sun gave sufficient conditions for the existence of infinitely many covering numbers, and he conjectured that these conditions were also necessary for a covering number to be primitive. Recently, the second author and Daniel White have shown that Sun's conjecture is false by finding infinitely many counterexamples. In this article, we give necessary and sufficient conditions for certain positive integers to be primitive covering numbers. We use these results to answer a question of Sun, and to prove the existence of infinitely many previously-unknown primitive covering numbers. We also show, for each of these new primitive covering numbers L  , that a covering can be constructed with distinct moduli using only a proper subset of the divisors d>1d>1 of L as moduli.