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 $\frac$ 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), who used the term block divisor.

Example

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

On the contrary, 6 is a divisor but not a unitary divisor of 60, as 6 and $\frac=10$ 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''):

:$\sigma_k^*(n) = \sum_ \!\! d^k.$

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 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
:$\frac = \sum_\frac.$

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
:$a\oplus b = \frac,\qquad a\odot b=(a,b)$
where $(a,b)$ denotes the greatest common divisor of ''a'' and ''b''.

Odd unitary divisors

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

:$\sigma_k^(n) = \sum_ \!\! d^k.$

It is also multiplicative, with Dirichlet generating function

:$\frac = \sum_\frac.$

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). he 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 $A \log x$ whereIvić (1985) p.395

:$A = \prod_p\left(\right) \ .$

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.Sandor et al (2006) p.115

OEIS sequences

* is σ^*₀(''n'')
* is σ^*₁(''n'') 
* to are σ^*₂(''n'') to σ^*₈(''n'') 
* is $2^\omega(n)$, the number of unitary divisors
* is σ^(o)*₀(''n'') 
* is σ^(o)*₁(''n'') 
* is $\sum_^\sigma_(i)$
* is $A$

References

* Section B3.
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* Section 4.2
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External links

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Mathoverflow , Boolean ring of unitary divisors

{{Divisor classes

Number theory