Smarandache Sequence And Function Of A Number Of Issues To Study

The research on the properties of arithmetical sequences is the kernel of the number theory. In 1993, Florentin Smarandache-the renowned American-Romanian theorist on number, proposed 105 unsolved mathematical problems and conjectures about special sequences and arithmetical functions in his book named "Only Problem, No Solution Answer!". With the presentation of these problems, Many number theory enthusiasts make great progress in some impor-tant unsolved issues and get a lot of theoretical results of great value, which contribute to the further development of number theory.Based on our interests in the above problems, we put our hands to study-ing the arithmetical properties of Smarandache function and special sequences which have not yet been resolved by using elementary and analytical method. Specifically, achievements made in this dissertation set as follows:1. Study the distribution properties of Smarandache 3n sequence and Smarandache 5n sequence; discuss the convergence and divergence of infinite series involving Smarandache 5n sequence and prove that if n take some special values, there is no perfect square in Smarandache 5n sequence.2. Define a new arithmetic function J(n) whose distribution property is studied in simple number set by using analytical way and finally a strong asymp-totic formula is given. Where J(n) is defined as follows:If n=p1α1p2α2…Pkαk (where pi is a prime,1≤i≤k) is the standard factor decomposition of positive integer n andβ≥1 is any given real number, then3. Discuss the hybrid mean value of Sdf(n) with P(n), and a asymptotic formula is given by analytical method at last.where Sdf (n) is Smarandache double factorial function, P (n) is the greatest prime factor of positive integer n.4.Discuss the solvability of equationφ(n)=S(n5) by classification way and give all solutions satisfying this equation.where S(n) is the Smarandache function,φ(n) is the Euler function.