Several Problems About Smarandache Sequence And Function

Professor Florentin Smarandache, who is the famous American-Romanian number theorist, put forward many problems about number theory and con-jecture in his book named "Only problem, Not solutions!" which published by the American Research Press in1993. This provoked people’s interests for the study of number theory. In addition, the famous Japanese number theory expert Dr. Kcnichrio Kashihara also proposed many problems associated with number theory. Many experts and scholars have made deep exploration and research on these issues and obtained a lot of achievements with important academic value.Based on the interests of the above issues, we will study several unresolved problems about Smarandache function and special sequence by using elementary and analytical method, and achieved some important valued results. Specifically, the four aspects including in the content arc as follows:1. Define a new Smarandache sequence, study its mean value properties and give several interesting mean value formulae and their applications.2. Study the hybird mean value properties of the Smarandache κn-digital sequence with the Smarandache function S(n) and the Smarandache power func-tion SP(n), and give some asymptotic formulae.3. Discuss the solvability of an equation involving the Smarandachc dual function(?) when k=4, and obtain its all positive intege solutions.4. The elementary methods are used to study the low bound estimate of the Smarandachc function, and get a group of asymptotic formulae.