The behavior of DS-divisors of positive integers

We study the behavior of DS-divisors of positive integers. Here "DS" stands for "divisor-squared." For an integer n, a positive divisor q of n is called a DS-divisor if q² | n - q. Such a pair (n, q) is called a DS-pair. Using a table generated for DS-pairs, we examine the existence and the numbers of positive DS-divisors of prime powers, products of two prime powers, and other cases represented by primary factorization. We also investigate patterns and structures of DS-divisors derived from our observations of the table. In addition, we study relationships between the numbers of DS-divisors and the values of Euler function. This research is related to the Primality Test problem of positive integers.