Smarandache Function Studies On Two Special Properties

Number theory, which is called the crown of mathematics, is an important branch of mathematics. Since Quadratic reciprocity law which is the the yeast of Mathematics was discovered in the eighteenth century, Number theory was familiar with more and more people. A large number of scientists fascinated by the search of Number theory. It is widely used in every branch of mathematics and its method is used in many subjects.Doctor Kenichiro Kashihara firstly proposed the Cubic order sequence in his book that " Comments and topics on Smarandache notions and problems".For any positive integer n, let Cn=n3, the Cubic order sequence is given by{zn}:{zn} is the smallest positive integer solutions of the congruence equation czn/n=1{modcn+1)Charles Ashbacher、Amarnath Murthy firstly proposed F.Smarandache factor partitions in their book that "Generalized Partitions and New Ideas On Number Theory and Smarandache Sequences".For any positive integer n>1, if n=d1d2…dn, where di is the divisor of n. Then we called the n=d1d2…dn as a F.Smarandache factor partitions of n.Based on these notions, we searched F.Smarandache p-th order sequence and F.Smarandache factor partitions of two kinds of special positive integer order. The main results as follows:1.Based on the notion of Cubic order sequence, defined the5-th order se-quence and proved the two conclusions as Cubic order sequence.2.Extanded to the p-th order sequence, also discovered the two natures were right in it. So we obtained the properties in the general p-th order sequence. 3.By the search of recursion formula of the sequence F(1#n), we got the computational formula of it. At the same time, we defined a new function f(a1a…2am), which made the problem solved easily.4.We discussed the special positive integer pn, and a perfect formula was given.