Suppose a sequence is defined by $u(1)=1$ and $u(n)= (n-1)u(n-1)+1$.
When does $n$ divide $u(n)$?
Numerical evidence shows that $1,2,4,5,10,13$ work. The only odd primes seem to be $5,13$ and $37$. The answer is probably all the divisors of $9620$, but I don't know how to prove this. $u(n)$ is $\displaystyle(n-1)!\sum_{k=0}^{n-1}\frac{1}{k!}$ in closed form, but that doesn't seem to achieve anything.
Any guidance would be much appreciated. Thanks.