The Nature Of The Equation Solution Of The Smarandache Function And The Pseudo-even Number Sequence

It is a very important content of elementary number theory to study the various properties of the arithmetical functions, and the famous Smarandache function S(n) is one of the important arithmetical functions, it is first introduced by American Romania famous mathematician Florentin Smarandache. In addition, in his book《Only problems, Not solusions》, Professor F.Smarandache proposed 105 special sequences, arithmetic functions and some unmsolved problems and conjectures. After that, many scholars had studied these problems, and obtained a series of very important research results.In this paper, based on the above mentioned problems and conjectures, we studied the various properties of some Smarandache sequences, and the solutions of some equations involving the Smarandache function. The main results as follows:1. The solutions of the equation (?)(mod n) has been studied, and its all positive integer solutions are given, if n is a square-free number.2. Using the elementary and combinatorial methods to study some calculating problems of the first and second Smarandache pseudo-even sequences, and two more sharper asymptotic formulae are given.3. Using the elementary method to study the properties of the Smarandache 3n-digital sequence, and part of the conjecture proposed by Professor Zhang Wenpeng. is solved.