Orders of Groups Characterised by Their Power Graphs

‎The power graph $P(G)$ of a group $G$ is the graph whose vertex‎ ‎set is the group elements and two elements are adjacent if one is‎ ‎a power of the other‎. ‎‎‎‎‎‎‎‎‎In this paper‎, ‎we raise and study the following question‎: ‎For which natural numbers $n$ every two groups of order $n$ with isomorphic power graphs are isomorphic?‎ ‎In particular‎, ‎it is proved that all such odd number $n$ are cube-free and also they are not multiples of $16$ in general‎.