TITLE: Are there infinitely many insipid numbers?
QUESTION [6 upvotes]: A number $n$ is called insipid if the groups having a core-free maximal subgroup of index $n$ are exactly $A_n$ and $S_n$. There is an OEIS enter for these numbers: A102842. There are exactly $486$ insipid numbers less than $1000$.
Question: Are there infinitely many insipid numbers? 
Let $\iota(r)$ be the number of insipid numbers less than $r$. The following plot (from OEIS) leads to:    
Bonus question: Is it true that $\lim_{r \to \infty}r/\iota(r)=2$?

REPLY [17 votes]: Almost all $n$ are insipid. In fact, the number of non-insipid numbers at most $n$ grows like $2n/\log n$. See the paper
Cameron, Peter J.; Neumann, Peter M.; Teague, David N.
On the degrees of primitive permutation groups.
Math. Z. 180 (1982), 141–149.
doi.org/10.1007/BF01318900