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. Thus, 5 is a unitary divisor of 60, because 5 and $\frac=12$ have only 1 as a common factor, while 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. 1 is a unitary divisor of every natural number.

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 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''):

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

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 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.

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. 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

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