  A positive integer n is called a covering number if there are some distinct
divisors n_1,...,n_k of n greater than one and some integers a_1,...,a_k such
that Z is the union of the residue classes a_1(mod n_1),...,a_k(mod n_k). A
covering number is said to be primitive if none of its proper divisors is a
covering number. In this paper we give some sufficient conditions for n to be a
(primitive) covering number; in particular, we show that for any r=2,3,...
there are infinitely many primitive covering numbers having exactly r distinct
prime divisors. In 1980 P. Erdos asked whether there are infinitely many
positive integers n such that among the subsets of D_n={d>1: d|n} only D_n can
be the set of all the moduli in a cover of Z with distinct moduli; we answer
this question affirmatively. We also conjecture that any primitive covering
number must have a prime factorization p_1^{alpha_1}...p_r^{alpha_r} (with
p_1,...,p_r in a suitable order) which satisfies $\prod_{0<t<s}(alpha_t+1)\ge
p_s-1$ for each s=1,...,r, with strict inequality when s=r.
