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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 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**.

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.

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.

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*^{rp} 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*^{rp} 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

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 denotes the greatest common divisor of *a* and *b*. ^{[2]}

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

It is also multiplicative, with Dirichlet generating function

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 where^{[3]}

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: 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 , the number of unitary divisors
- OEIS: A068068 is σ
^{(o)*}_{0}(*n*) - OEIS: A192066 is σ
^{(o)*}_{1}(*n*) - OEIS: A064609 is
- OEIS: A306071 is

**^**R. Vaidyanathaswamy (1931). "The theory of multiplicative arithmetic functions".*Transactions of the American Mathematical Society*.**33**(2): 579–662. doi:10.1090/S0002-9947-1931-1501607-1.**^**Conway, J.H.; Norton, S.P. (1979). "Monstrous Moonshine".*Bulletin of the London Mathematical Society*.**11**(3): 308–339.**^**Ivić (1985) p.395**^**Sandor et al (2006) p.115

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*MathWorld*. - Mathoverflow | Boolean ring of unitary divisors