Find a linear bounded automaton that accepts the language $L = \{ a^{n!} : n \geq 0 \}$
The LBA maintains a counter $c$, initialized by $1$, which is stored on a parallel track. It thinks of the rest of the tape as an integer $m$. It then repeatedly executes the following instructions: divide $m$ by $c$, and increment $c$. The algorithm terminates when one of the following happens: $m = 1$, in which case you can declare success; or $m$ is not divisible by $c$, in which case you can declare failure.