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

let $gamma(s_n)$ be the minimum number of proper subgroups‎ ‎$h_i, i=1‎, ‎dots‎, ‎l $ 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 $ninmathbb{n}$ 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 ‎{tt magma} computation‎. ‎we make a similar conjecture for $gamma(a_n)$‎, ‎when $n$ is even‎, ‎and provide a similar amount of evidence‎.