New York Journal of Mathematics
Let φ(x) = x d + c be an integral polynomial of degree at least 2, and consider the sequence (φ n (0))∞n=0, which is the orbit of 0 under iteration by φ. Let Dd,c denote the set of positive integers n for which n | φ n (0). We give a characterization of Dd,c in terms of a directed graph and describe a number of its properties, including its cardinality and the primes contained therein. In particular, we study the question of which primes p have the property that the orbit of 0 is a single p-cycle modulo p. We show that the set of such primes is finite when d is even, and conjecture that it is infinite when d is odd.
Chen, Annie S.; Gassert, T. Alden; and Stange, Katherine E., "Index divisibility in dynamical sequences and cyclic orbits modulo p" (2017). Mathematics Faculty Contributions. 4.