Strictly sharp irreducible characters of symmetric and alternating groups

A complex character $\chi$ of a finite group $G$ is called strictly sharp if $|G|=\prod_{l\in L}(n-l)$, where $n=\chi(1)$ is the degree of the character and $L=\{\chi(g)\mid g\in G,\ g\ne1\}$. In this paper all irreducible strictly sharp characters of the symmetric and alternating groups are found. In particular, it is proved that the symmetric groups $S_n$, $n\geslant7$, and the alternating groups $A_n$, $n\geslant9$, have exactly one irreducible strictly sharp character.