When is $u_n$ defined by $u_1=1,u_n=(n-1)u_{n-1}+1$ divisible by $n$?

Question

Problem statement : for which values of $n$ is $u_n$ defined by $u_1=1,u_n=(n-1)u_{n-1}+1$ divisible by $n$ ?

I have tried introducing an auxiliary recurrence relation like $v_n=u_n-1$ but this is not getting me anywhere.

I have also tried computing the first few values of $u_n$ but I cannot see a pattern.

Note : this is denoted as 'hard' in my text, 'The Mathematical Olympiad Handbook - an introduction to problem solving' by A.Gardiner.

Answer 1

This is likely a problem that has never been corrected; see this post for your exact problem, and check out the comments.