The characterization of some cyclic groups by the product of element orders

Let G be a finite group. We denote by psi '(G) the integer Pi(g is an element of G)o(g), where o(g) denotes the order of g is an element of G. It is verified that some finite groups are uniquely determined by their orders and their product of element orders. In this paper, we prove that, if p, q, r, t are different prime numbers, then Z(p3), Z(pq), Z(p2q) and Z(pqrt) are characterized by psi '(G) and if psi '(G) = psi '(Z(pqr)), then G congruent to A(5) or Z(pqr).