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. 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),^[1] who used the term block divisor.

Example

The integer 5 is a unitary divisor of 60, because 5 and ${\displaystyle {\frac {60}{5}}=12}$ have only 1 as a common factor.

On the contrary, 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.

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):

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

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 2^k, where k is the number of distinct prime factors of n. This is because each integer N > 1 is the product of positive powers p^r_p 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 p^r_p for p ∈ S. If there are k prime factors, then there are exactly 2^k 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.

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

${\displaystyle a\oplus b={\frac {ab}{(a,b)^{2}}},\qquad a\odot b=(a,b)}$

where ${\displaystyle (a,b)}$ denotes the greatest common divisor of a and b. ^[2]

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. 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 ${\displaystyle A\log x}$ where^[3]

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

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.^[4]

OEIS sequences

• OEIS: A034444 is σ^*₀(n)
• OEIS: A034448 is σ^*₁(n) 
• OEIS: A034676 to OEIS: A034682 are σ^*₂(n) to σ^*₈(n) 
• OEIS: A034444 is ${\displaystyle 2^{\omega }(n)}$, the number of unitary divisors
• OEIS: A068068 is σ^(o)*₀(n) 
• OEIS: A192066 is σ^(o)*₁(n) 
• OEIS: A064609 is ${\displaystyle \sum _{i=1}^{n}\sigma _{1}(i)}$
• OEIS: A306071 is ${\displaystyle A}$