Researches On Properties Of The Smarandache Function S(n)

For any positive integer n, the famous Smarandache function S(n) is defined as the smallest positive integer m and n|m!, that is S(n) = min{m:n|m!}. Let n be any positive integer, if the product of all proper factors of n is less than or equal to n, then n is said a simple number.There are five chapters in this thesis.In Chapter 1, the background and the significance of research, include the research progress about the Smarandache function S(n), the sequence of simple numbers and primitive number of power p. The main results and the content organization of this thesis are also given.In Chapter 2, preliminaries related to the research, include several classical arith-metic functions and the famous Smarandache function S(n), Abel identity, and Euler summation formula.In Chapter 3, we consider the mean value properties of function S(n) about the sequence of simple numbers by using elementary method, and two interesting asymptotic formulas are given.In Chapter 4, we investigate the mean value properties of arithmetic functions that contain function S(n) about the sequence of simple numbers, include several classical arithmetic functions and other famous Smarandache function.In Chapter 5, we consider some properties of primitive number of power p. From the known relationship between function Sp(n2) and Sp(n), further promote to the relation-ship between function Sp(nk) and Sp(n), the correspondence between the values of Sp(n) and the number of items is also studied.