I've been trying an old question from Gardiner's excellent Mathematical Olympiad Handbook which drove me mad a few years ago and continues to do so now!
Consider the recurrence relation
$$u_n = (n-1)u_{n-1} + 1 \, , \quad u_1 = 1$$
When is $u_n$ divisible by $n$?
It's not hard to show that
$$u_n = \sum_{k=0}^{n-1} \frac{(n-1)!}{k!} $$
which is exactly sequence A000522. However this observation doesn't seem to help a huge amount.
The first few $n$ for which the criterion are satisfied are
$$n=1,2,4,5,10,13,16$$
which doesn't lend itself to an obvious conjecture!
Does anyone have an inkling as to how to attack this problem? Hints very welcome!