On the factorization numbers of some finite $p$-groups

This note deals with the computation of the factorization number $F_2(G)$ of a finite group $G$. By using the M"{o}bius inversion formula, explicit expressions of $F_2(G)$ are obtained for two classes of finite abelian groups, improving the results of {\it Factorization numbers of some finite groups}, Glasgow Math. J. (2012).