Several New Smarandache Sequence Of The Nature Of The Research

Number theory is closely related to the development of mathematics. It has been an important issue to the research on the properties of some arith-metical functions and sequences. Florentin Smarandache, who is a a renowned American-Romanian theorist on number in his "only problem, there is no solu-tion A!" mentioned some unresolved problems of number theory and conjectures. These problems contain some interests and skills. The resolution of these issues has been greatly promoted the development of the interest and skills, provid-ing a broad space to thinking for the researchers and inspiring their questing for knowledge. They have got some theoretical results of important value, which contributing to the immeasurable effect about the constant development of num-ber theory.I get some meaningful conclusions to the unresolved problems by using the elementary and analytical methods. As following:(1)The properties of the Smarandache summation functions;(2) The mean value distribution properties of the new Smarandache number sequences;(3) Investigating the mean problem of the Smarandache repeating number sequence.1. Give the new definition of Smarandache summation functions, at the same time, studying the Dirichlet series which involving Smarandache sum-mation functions, finally, getting some useful equations. The definition of this function as following:to any positive integer n and k>1,2. Get two useful asymptotic formula by studying mean value property of Smarandache kn number sequence on the basis of extending the definition of Smarandache3n.3. Discuss the asymptotic property of the adjacent number sequence b(n, m).4. Studying the mean value of Smarandache repeated natural number series by using the analytical methods and get a meaningful conclusion.