On The Equation Of The Smarandache Function And An Important Class Of Sequence Nature

It is a hot topic to study arithmetical functions and sequences in number theory. American-Romanian number theorist F. Smarandache presented 105 unsolved problems in a book named "Only problems, Not solutions!". These problems were investigated by many researchers, and numerous valued results were obtained.Based on the interests in Smarandache functions and Smarandache sequences, in this thesis we investigate the solvability of some equations which are composed of Smarandache functions, pseudo-Smarandache functions and Euler functions, we also explore the divisibility for Smarandache sequences. The main achievements are as follows:1. The number and form of the solutions for three equations composed of Smarandache functions and Euler functions are obtained, and a related conjecture is proposed.2. All positive integer solutions for two equations composed of Smarandache functions and pseudo-Smarandache functions are given.3. Some divisibility properties of Smarandache numerical carpet sequences are established.