CN-TST 2012 Problem 12

Given an integer $n\ge 2$ , a function $f:\mathbb{Z}\rightarrow \{1,2,\ldots,n\}$ is called good, if for any integer $k,1\le k\le n-1$ there exists an integer $j(k)$ such that for every integer $m$ we have $$f(m+j(k))\equiv f(m+k)-f(m) \pmod{n+1}.$$ Find the number of good functions.