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 \frac{b}{a} are coprime, having no common factor other than 1. 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 concept of a unitary divisor originates from R. Vaidyanathaswamy (1931), who used the term block divisor.

Example

The integer 5 is a unitary divisor of 60, because 5 and have only 1 as a common factor. On the contrary, 6 is a divisor but not a unitary divisor of 60, as 6 and have a common factor other than 1, namely 2.

Sum of unitary divisors

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): It is a multiplicative function. If the proper unitary divisors of a given number add up to that number, then that number is called a unitary perfect number.

Properties

Number 1 is a unitary divisor of every natural number. 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 set of all unitary divisors of n forms a Boolean algebra with meet given by the greatest common divisor and join by the least common multiple. Equivalently, the set of unitary divisors of n forms a Boolean ring, where the addition and multiplication are given by where (a,b) denotes the greatest common divisor of a and b.

Odd unitary divisors

The sum of the k-th powers of the odd unitary divisors is It is also multiplicative, with Dirichlet generating function

Bi-unitary divisors

A divisor d of n is a bi-unitary divisor if the greatest common unitary divisor of d and n/d is 1. This concept originates from D. Suryanarayana (1972). [The number of bi-unitary divisors of an integer, in The Theory of Arithmetic Functions, Lecture Notes in Mathematics 251: 273–282, New York, Springer–Verlag]. The number of bi-unitary divisors of n is a multiplicative function of n with average order A \log x where A bi-unitary perfect number is one equal to the sum of its bi-unitary aliquot divisors. The only such numbers are 6, 60 and 90.

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