Smarandache Sequence And Function Of The Mean

Number Theory is a subject about integer properties,which plays a more important role in Mathematics.In Number Theory,the mean value properties of some special sequences and functions have been researched by many mathematicians and scholars,which related to many famous number conjectures and unsolved problems.Therefore,any nontrivial progress in the fields will contribute to the development of elementary number theory and analytic number theory.Number theorist Florentin Smarandache has presented many problems and conjectures on special sequences and arithmetical functions.He published a book named《only problems, not solutions》in American in 1993.He presented 105 unsolved arithmetical problems and conjectures on special sequences and arithmetical functions.Many researchers have studied these sequences and functions from this book,and obtained some important valued results on the theory.Doctor Kenichiro Kashihara presented many problems on Number Theory involving Smarandache functions in《comments and topic on Smarandache notions and problems》,Some of them have much value and been researched by many people.In this paper,we use elementary and analytic methods to study some problems on some special sequences and functions,and give their mean values,concisely counting formula and asymptotic formulae.The main achievements contained in this dissertation are as follows:1.Studying the mean value of some new Smarandache digital sums functions,and giving some calculating formulas about them.2.Giving the mean value of some special functions on Smarandache Fibonacci Base.3.Studying the mean value of two Smarandache LCM Functions by using elementary and analytic method,and giving their asymptotic formula.