All Random Solved Random Open
Let $A$ be the set of $n\in \mathbb{N}$ such that there exist $1\leq m_1<\cdots <m_k=n$ with $\sum\tfrac{1}{m_i}=1$. Explore $A$. In particular,
  • Does $A$ have density $1$?
  • What are those $n\in A$ not divisible by any $d\in A$ with $1<d<n$?
Straus observed that $A$ is closed under multiplication. Furthermore, it is easy to see that $A$ does not contain any prime power.
Additional thanks to: Zachary Hunter