Smarandache Function And The Column Equation Solver And Mean

It is known that the research on the properties of all kinds of arithmetical sequences play an important role in the number theory, and they related to many famous number theoretic problems. Therefore, any nontrivial progress in this field will propel the development of elementary number theory and analytic number theory.American-Romanian number theorist Florentin Smarandache presented many problems and conjectures on Smarandache functions. In 1993. he published a book named "Only problems, Not solutions!" in Xiquan Publishing House in American. In this book, he presented 105 unsolved arithmetical problems and conjectures about special sequences and functions, many researchers studied these problems. And Kenichiro Kashihara doctor in Japan also proposed a lot of number theory problems about Smarandache functions in the book named " Comments and Topics On Smarandache Notions and Problems " , some problems have good values and inspire the readers interesting.In this dissertation, we used elementary methods and analytical methods to study some new Smarandache functions, sequences and near pseudo Smarandache function, and given some related identities, asymptotic formulas and positive integer solutions of some equations. The main achivements contained in this dissertation are as follows:1. We defined two arithmetical functions and got mean value properties of the two functions by using elementary and analytic methods; we also studied the mean value property of Smarandache ceil function S_k(n!) and given an asymptotic formula.2. Smarandache LCM dual function SL*(n) and Smarandache multiplica- tive function SM(n) have important positions in the study of elementary number theory. We used elementary method to study the positive solutions of some equations about them respectively.3. It is quite meaningful to research the near pseudo Smarandache function. We studied the properties of near pseudo Smarandache function by using elementary method and obtained some interesting identities, at the same time, we presented an open problem.4. We defined a new Smarandache function G(n) and studied its elementary properties by using elementary method.