Criteria for the integrality of nth roots of power series

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