Let $T(x)=\Sigma\tau(G)$, where $\tau(G)$ denotes of direct factors of an Abelian group $G$. It is know that $T(x)=\sum_{n\leq x}t(n)$ where $t(n)$ is a multiplicative function such that $\sum^\infty_{n=1}t(n)n^{-s}=\zeta^2(s)\zeta^2(2s)\zeta^2(3s)\dots(\operatorname{Re} s>1)$, and $\zeta(s)$ as usual denotes the Riemann zata-function. The aim of this note is to investigate some asymptotic formulas for the summatory functions of $t^k(n)$.