A Class Of Special Collections In Nature

Number theory is an important branch of mathematics. It is widely used in every branch of mathematics and its method is used in many subjects. Amarnath Murthy and Charles Ashbacher proposed co-prime sets in their book that called "Generalized Partitions and New Ideas On Number Theory and Smarandache Sequences". For any positive integer n, let α1,α2,α3,…, αk are all positive divisors of n. All pairs of co-prime divisors are chosen to be a set. The set is called as a pairwise co-prime set D2(n). All three of divisors which are co-prime are chosen to be a set. The set is called as a triplet co-prime set D3(n). All r pairs of divisors are chosen to be a set. This set is called as a r co-prime set Dr(n). Amarnath Murthy and Charles Ashbacher had studied the arithmetical properties of Dr(n). At the same time, they also proposed a series of problems and conjectures related to Dr(n).Based on that concept, the main results as follows:1. The elementary methods are used to give an exact computation formula for|D2(n)|and|D3(n)|, for any integer n, the computation formula are as follows: the d(n) is the Dirichlct divisor functions.2. We proposed new concept when we proved the theory and find the relationship of the r co-prime sets And we find new conjecture based on that relationship and the computation:if |Dr1(n)|is then xi is the result of an equation. It is proved in the chapter4.3. Based on previous proof, there is a new conjecture that the computation formula of orders of r co-prime sets is then yi is the result of an equation. It is also proved in the chapter4.