# Unitary divisor

In mathematics, a natural number *a* is a **unitary 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 given 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.

## Properties

The number of unitary divisors of a number *n* is 2^{k}, where *k* is the number of distinct prime factors of *n*. 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

## 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. The number of bi-unitary divisors of *n* is a multiplicative function of *n* with average order where^{[1]}

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.^{[2]}

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