The possible numbers are $a_n = \frac{10^n - 1}{9}$, for $2 \leq n \leq 10$. If $n > 2$ and $2 | n$, then $11 |a_ n$, and if $3 | n$, then $3 | 111 |a_n$. This rules out $ n = 3, 4, 6, 8, 9, 10$. This leaves only 2, 5, and 7, so we are done.