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

The web of Science code is WOS:000215362600006. The system does not allow to insert it. 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)=n/2(1-1/p_1) (1-1/p_2)+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) different from (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.