# 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 ${\displaystyle {\frac {b}{a}}}$ are coprime, having no common factor other than 1. Thus, 5 is a unitary divisor of 60, because 5 and ${\displaystyle {\frac {60}{5}}=12}$ have only 1 as a common factor, while 6 is a divisor but not a unitary divisor of 60, as 6 and ${\displaystyle {\frac {60}{6}}=10}$ 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):

${\displaystyle \sigma _{k}^{*}(n)=\sum _{d\mid n \atop \gcd(d,n/d)=1}\!\!d^{k}.}$

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 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 divisors, 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

${\displaystyle {\frac {\zeta (s)\zeta (s-k)}{\zeta (2s-k)}}=\sum _{n\geq 1}{\frac {\sigma _{k}^{*}(n)}{n^{s}}}.}$

Every divisor of n is unitary if and only if n is square-free.

## Odd unitary divisors

The sum of the k-th powers of the odd unitary divisors is

${\displaystyle \sigma _{k}^{(o)*}(n)=\sum _{{d\mid n \atop d\equiv 1{\pmod {2}}} \atop \gcd(d,n/d)=1}\!\!d^{k}.}$

It is also multiplicative, with Dirichlet generating function

${\displaystyle {\frac {\zeta (s)\zeta (s-k)(1-2^{k-s})}{\zeta (2s-k)(1-2^{k-2s})}}=\sum _{n\geq 1}{\frac {\sigma _{k}^{(o)*}(n)}{n^{s}}}.}$

## 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 ${\displaystyle A\log x}$ where[1]

${\displaystyle A=\prod _{p}\left({1-{\frac {p-1}{p^{2}(p+1)}}}\right)\ .}$

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]

## OEIS sequences

• is σ0(n
•   is σ1(n
•   to are σ2(n) to σ8(n
•   is σ(o)*0(n
•   is σ(o)*1(n
•   is ${\displaystyle \sum _{i=1}^{n}\sigma _{1}(i)}$;

## References

1. ^ Ivić (1985) p.395
2. ^ Sandor et al (2006) p.115