On covering numbers

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A positive integer n is called a covering number if there are some distinct divisors n₁,...,nₖ of n greater than one and some integers a₁,...,aₖ such that Z is the union of the residue classes a₁(mod n₁),...,aₖ(mod nₖ). 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 Dₙ=d>1: d|n only Dₙ 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 p₁alpha₁...pᵣalphaᵣ (with p₁,...,pᵣ in a suitable order) which satisfies ∏₀<ₜ<ₛ(alphaₜ+1)≥ pₛ-1 for each s=1,...,r, with strict inequality when s=r.