  We prove a new criterion for the solvability of the finite groups, depending
on the function $\psi_k(G)$ which is defined as the sum of $k$-th powers of the
element orders of $G$. We show that our result can be used to show the
solvability of some groups for which the solvability does not follow from
earlier similar kind of results and we emphasize the following: looking at
$\psi_k(G)$ for $k>1$ can be useful to get further pieces of information about
the group $G$.
