About The Smarandache Function Equation Solvability Research And Mean Estimates

As we all know, number theory is the branch of mathematics which studies the nature of integer. Both properties of arithmetic functions and the mean value of special sequences are important research topics in number theory. The famous number theory expert named Florentin Smarandache whose book "Only the problems, Not solutions" has put forward105about arithmetic functions and special sequences which hadn’t been solved, known as Smarandache problems. These problems caused many arithmetic scholar’s research interests.Based on the interest of Smarandache problems and the current research situation, part of the problems were discussed and relevant results are obtained in this paper as follows:1. By using the elementary methods and classification methods to study the equation which containing Smarandache LCM dual function SL*(n) and prime factor function Ω(n), and all the positive integer solutions are obtained when k is equal to1and2.2. The above equation is generalized and part of positive integer solutions of the equation are obtained.3. Studying the positive integer solutions of equation S5(1)+S5(2)+…+S5(n)=5Ω(n) which involving pseudo-Smarandache Ceil function Sk(n).4. Using the analytic methods to estimate the mean value of Smaran-dache power function SP(n) on simple data set, and the asymptotic formula ofare given.