Inmathematics, anatural numbera is aunitary divisor (orHall divisor) of a numberb ifa is adivisor ofb and ifa and${\displaystyle \frac{b}{a}}$ arecoprime, having no common factor other than 1. Equivalently, a divisora ofb is a unitary divisorif and only if everyprime factor ofa has the samemultiplicity ina as it has inb.

The concept of a unitary divisor originates from R. Vaidyanathaswamy (1931),^[1] who used the termblock divisor.

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.

The sum-of-unitary-divisors function is denoted by the lowercase Greek letter sigma thus: σ*(n). The sum of thek-thpowers of the unitary divisors is denoted by σ*_k(n):

${\displaystyle {\sigma }_k^{\ast }(n)=\sum _{\genfrac{}{}{0ex}{}{d\mid n}{gcd(d,n/d)=1}}{d}^k.}$

It is amultiplicative function. If theproper unitary divisors of a given number add up to that number, then that number is called aunitary perfect number.

Number 1 is a unitary divisor of every natural number.

The number of unitary divisors of a numbern is 2^k, wherek is the number of distinct prime factors ofn. This is because eachintegerN > 1 is the product of positive powersp^r_p of distinct prime numbersp. Thus every unitary divisor ofN is the product, over a given subsetS of the prime divisors {p} ofN,of the prime powersp^r_p forp ∈S. If there arek prime factors, then there are exactly 2^k subsetsS, and the statement follows.

The sum of the unitary divisors ofn isodd ifn is apower of 2 (including 1), andeven otherwise.

Both the count and the sum of the unitary divisors ofn aremultiplicative functions ofn that are notcompletely multiplicative. TheDirichlet generating function is

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

Every divisor ofn is unitary if and only ifn issquare-free.

The set of all unitary divisors ofn forms aBoolean algebra with meet given by thegreatest common divisor and join by theleast common multiple. Equivalently, the set of unitary divisors ofn forms a Boolean ring, where the addition and multiplication are given by

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

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

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

${\displaystyle {\sigma }_k^{(o)\ast }(n)=\sum _{\genfrac{}{}{0ex}{}{\genfrac{}{}{0ex}{}{d\mid n}{d\equiv 1(\mathrm{mod}2)}}{gcd(d,n/d)=1}}{d}^k.}$

It is also multiplicative, with Dirichlet generating function

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

A divisord ofn is abi-unitary divisor if the greatest common unitary divisor ofd andn/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 ofn is a multiplicative function ofn withaverage order${\displaystyle A\log x}$ where^[3]

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

Abi-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: A034444 is σ^*₀(n)
• OEIS: A034448 is σ^*₁(n) 
• OEIS: A034676 toOEIS: 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${\displaystyle A}$

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• Mathoverflow | Boolean ring of unitary divisors

• Number theory

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