On the difference between the mean value of the Smarandache square parts sequences SP(n) and IP(n)

For any positive integer n,the Smarandache Superior Square Part SP(n) is the smallest square greater than or equal to n,the Smarandache Inferior Square Part IP(n) is the largest square less than or equal to n.Japanese scholar asked us to study several mean value problems related to sequences SP(n) and IP(n). Recently,Chinese scholar first used the elementary and analytic methods to study these problems,obtained several interesting mean value formulae,and solved several problems proposed by Japanese scholar.The main purpose of this paper is also to study the properties of the Smarandache square parts sequences,and give a new asymptotic formula involving SP(n) and IP(n).