A new characterization of alternating groups

Let $G$ be a finite group and let $\pi_{e}(G)$ be the set of element orders of $G $. Let $k \in \pi_{e}(G)$ and let $m_{k}$ be the number of elements of order $k $ in $G$. Set $\mathrm{nse}(G):=\{ m_{k} | k \in \pi_{e}(G)\}$. In this paper, we show that if $n = r$, $r +1 $, $r + 2$, $r + 3$ $r+4$, or $r + 5$ where $r\geq5$ is the greatest prime not exceeding $n$, then $A_{n}$ characterizable by nse and order.