For a finite group $G$, $\#$Cent($G)$ denotes the number of centralizers of its elements. A group $G$ is called $n$-centralizer if $\#$Cent($G) = n$, and primitive $n$-centralizer if $\#$Cent($G) = \#$Cent($\frac{G}{Z(G)}) = n$.
In this paper we compute the number of distinct centralizers of some finite groups and prove that for any positive integer $n \ne 2, 3$, there exists a finite group $G$ with $\#$Cent($G) = n$, a question raised by Belcastro and Sherman [2]. Also, we investigate the structure of finite groups with exactly six distinct centralizers and prove that if $G$ is a primitive 6-centralizer group then $\frac{G}{Z(G)} \cong A_4$, the alternating group on four symbols. Furtheremore, we prove that if $\frac{G}{Z(G)} \cong A_4$, then $\#$Cent($G) = 6$ or 8 and construct a group with $\frac{G}{Z(G)} \cong A_4$ and $\#$Cent($G$) = 8. Finally, we prove some results about primitive 7-centralizer groups.\\ {\bf 1991 Mathematics Subject Classification:} 20D99, 20E07 \\ {\bf Key Words:} Centralizer, $n$-centralizer, primitive $n$-centralizer