  Denote the sum of element orders in a finite group $G$ by $\psi(G)$ and let
$C_n$ denote the cyclic group of order $n$. In this paper, we prove that if
$|G|=n$ and $\psi(G)>\frac{13}{21}\,\psi(C_n)$, then $G$ is nilpotent.
Moreover, we have $\psi(G)=\frac{13}{21}\,\psi(C_n)$ if and only if $n=6m$ with
$(6,m)=1$ and $G\cong S_3\times C_m$. Two interesting consequences of this
result are also presented.
