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Let $\gamma(S_n)$ be the minimum number of proper subgroups $H_i$ of thesymmetric group $S_n$ such that each element in $S_n$ lies in some conjugate ofone of the $H_i.$ In this paper we conjecture that$$\gamma(S_n)=\frac{n}{2}\left(1-\frac{1}{p_1}\right)\left(1-\frac{1}{p_2}\right)+2,$$ where $p_1,p_2$ are the two smallest primesin the factorization of $n$ and $n$ is neither a prime power nor a product oftwo primes. Support for the conjecture is given by a previous result for$n=p_1^{\alpha_1}p_2^{\alpha_2},$ with $(\alpha_1,\alpha_2)\neq (1,1)$. We givefurther evidence by confirming the conjecture for integers of the form $n=15q$for an infinite set of primes $q$, and by reporting on a Magma computation. Wemake a similar conjecture for $\gamma(A_n)$, when $n$ is even, and provide asimilar amount of evidence.

Conjectures on the normal covering number of the finite symmetric and alternating groups