  Let $\gamma(S_n)$ be the minimum number of proper subgroups $H_i$ of the
symmetric group $S_n$ such that each element in $S_n$ lies in some conjugate of
one 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 primes
in the factorization of $n$ and $n$ is neither a prime power nor a product of
two 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 give
further 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. We
make a similar conjecture for $\gamma(A_n)$, when $n$ is even, and provide a
similar amount of evidence.
