Some new problems about the Smarandache function and related problems

For any positive integer n, the famous F.Smarandache function S(n) is defined as
the smallest positive integer m such that njm!. That is, S(n) = min fm : m 2 N; njm!g.
The main purpose of this paper is to introduce some new unsolved problems involving the Smarandache function and the related functions.