On A Special Sequence Mean The Smarandache Function And The Related Nature

The study on the properties of all kinds of arithmetical functions and spe-cial sequences play a important role in the number theory. In 1993, Florentin Smarandache, who is a famous American-Romanian number theorist, published a book named "Only problem, Not solutions!" in Xiquan Publishing House in America. In this book, he defined many arithmetical functions and special se-quences, and proposed 105 unsolved problems and conjectures about those arith-metical functions and special sequences. Then many scholars made intensive studies of them, and obtained some important valued results on theory.In this dissertation, we used elementary methods and analytical methods to study some Smarandache functions, sequences and pseudo Smarandache func-tion, and given some asymptotic formulas and positive integer solutions of some equations. The main achievements contained in this dissertation are as follows:1. We studied the property of Smarandache function S(n), the Smaran-dache divisor product sequences {pd(n)} and the Smarandache proper divisor product sequences {qd(n)}. secondly, the elementary and analytic methods are used to study the mean value distribution property of S(n) on the Smaran-dache divisor product sequences and the Smarandache proper divisor product sequences, and two interesting asymptotic formula is given for it.2. The elementary and analytic methods are used to study the mean value distribution property of SM(n) on the Smarandache divisor product sequences and the Smarandache proper divisor product sequences, Then an interesting asymptotic formula is given for it. And the mean value distribution property of SM(n) on L(n) was studied also.3. Using the elementary method to study the solutions of the equation Z*(n)=SL(n), and give its all positive integer solutions.