On Smarandache Problems, Several New Arithmetic Function And Its Mean Value,

American Romania famous mathematician Professor Florentin Smaran-dache proposed many theories and practical problems, mostly contents related to number theory. Some scholars had studied these theoretical problems, and obtained a series of important research results, they published in some inter-national mathematical journals, such as "Smarandache Notions Journal" and "Scientia Magna. ". On the other hand, in his book "Comments and Topics On Smarandache Notions and Problems", Japanese mathematician Kashihara Kenichiro also proposed many problems related number theory and Smarandache functions, in which many problems have important theoretical significance and value. Based on these problems, we studied the solvability of some equations in-volving the pseudo Smarandache function and its dual function. We also defined two new arithmetical functions. Smarandache power summands functions. Then we studied the solvability of some equations and the properties of the Smaran-dache power summands functions. The main results including the following three aspects:1. Studied the solvability of some equation involving the pseudo Smaran-dache function and its dual function by using the elementary and combinational method. A series of positive integer solutions are given for the equation. Finally, we proved that the odd number n satisfying the equation if and only if n= pk, where p≥5 be a prime, and k be any positive integer.2. For any positive integer n and any fixed integer k＞1, we defined two Smarandache power summands functions P(n,k) and AP(n,k). Then we studied the mean value properties of these two functions, and give two interesting asymptotic formulae for them. 3.A new arithmetical function F(n) is defined by F(1)=0,and F(n)=α1p1+…+αkpk,if n=p1α1p2α2…pkαk is the prime power decomposition of n.Let P(n)be the largest prime divisor of n.In this part,we obtained an asymptotic formula for the mean value of(F(n)-P(n))2.