Unitary divisor

In mathematics, a natural number a is a unitary divisor (or Hall divisor) of a number b if a is a divisor of b and if a and are coprime, having no common factor other than 1. Thus, 5 is a unitary divisor of 60, because 5 and have only 1 as a common factor, while 6 is a divisor but not a unitary divisor of 60, as 6 and have a common factor other than 1, namely 2. 1 is a unitary divisor of every natural number. Equivalently, a divisor a of b is a unitary divisor if and only if every prime factor of a has the same multiplicity in a as it has in b. The sum-of-unitary-divisors function is denoted by the lowercase Greek letter sigma thus: σ*(n). The sum of the k-th powers of the unitary divisors is denoted by σ*k(n): If the proper unitary divisors of a given number add up to that number, then that number is called a unitary perfect number. The concept of a unitary divisor originates from R. Vaidyanathaswamy (1931) [The theory of multiplicative arithmetic functions. Transactions of the American Mathematical Society, 33(2), 579--662] who used the term block divisor. The number of unitary divisors of a number n is 2k, where k is the number of distinct prime factors of n. This is because each integer N > 1 is the product of positive powers prp of distinct prime numbers p. Thus every unitary divisor of N is the product, over a given subset S of the prime divisors {p} of N, of the prime powers prp for p ∈ S. If there are k prime factors, then there are exactly 2k subsets S, and the statement follows. The sum of the unitary divisors of n is odd if n is a power of 2 (including 1), and even otherwise. Both the count and the sum of the unitary divisors of n are multiplicative functions of n that are not completely multiplicative. The Dirichlet generating function is Every divisor of n is unitary if and only if n is square-free. The sum of the k-th powers of the odd unitary divisors is It is also multiplicative, with Dirichlet generating function A divisor d of n is a bi-unitary divisor if the greatest common unitary divisor of d and n/d is 1.