Question 144
Let $n$ be a positive integer not divisible by the cube of a prime. Given an integer-coefficient polynomial $f(x)$, define its signature modulo $n$ to be the (ordered) sequence $f(1), \dots, f(n)$ modulo $n$. Of the $n^n$ such $n$-term sequences of integers modulo $n$, how many are the signature of some polynomial $f(x)$?