Criteria for the integrality of $n$th roots of power series

Heninger, Rains and Sloane raised the question of which power series with integer coefficients can be written as the $n$th power of another power series with integer coefficients and constant term $1$. We provide necessary and sufficient conditions, as well as compare them with a general integrality criterion due to Dieudonné and Dwork that can be applied to this question as well.