The Nature Of Catalan Number As Well As The Study Of Several Equations Of Smarandache Function

Catalan number is an important counting function in Combination of count-ing theory, it is researched by qing dynasty mathemtician antu Ming firstly, and has attracted the attention of many experts and scholars. It has an wide appli-cation in combinatorical mathematics and graph theory, and has an application in solving the problem of Brackets, Stack order problem, the problem of the polygon is divided into a triangle and the problem of Composing a binary tree to the top node. Smarandache function is proposed firstly by the famous Amer-ican Romanian number theory experts F.Smarandache Professor who suggest that we study the relevant issues in the book 《Only problems,no solution!》 many domestic scholars have a lot of research and exploration of this topic, and achieved good results. Thus this paper include two aspects:On one hand, by using the elementary method and the properties of the Catalan numbers, we get five formulas and the results when k took special values from the generating function of Ctalan numbers.On the other hand, by using the properties of Pseudo-Smarandache func-tion and Euler function, we solve the solvability of four equations on Pseudo- Smarandache function and Euler function Z(n)=φ(n) and get all positive integer solutions.