On the Hybrid Mean Value of the Smarandache Double Factorial Function and the Pseudo-Smarandache Function

For any positive integer n,the famous Smarandache double factorial function Sdf( n) is defined as the smallest positive integer m such that n | m! !. That is Sdf( n) = min { m ∶m ∈ N,n | m! ! }. And the Pseudo-Smarandache function Z( n) is defined as the smallest positive integer m such that nm( m + 1) /2. That is,Z( n) = min{ m∶m∈N,nm( m + 1) /2}. The main purpose of this paper is to study the mean value properties of the composite function Sdf( Z( n)) and give a sharper asymptotic formula by the elementary method and analytic method.