# Are there infinitely many insipid numbers?

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 $$\lim_{r \to \infty}r/\iota(r)=2$$?

Almost all $$n$$ are insipid. In fact, the number of non-insipid numbers at most $$n$$ grows like $$2n/\log n$$. See the paper