Unitary divisor

Template:Short description In mathematics, a natural number Template:Mvar is a unitary divisor (or Hall divisor) of a number Template:Mvar if Template:Mvar is a divisor of Template:Mvar and if Template:Mvar and Template:Mvar are coprime, having no common factor other than 1. Equivalently, a divisor Template:Mvar of Template:Mvar is a unitary divisor if and only if every prime factor of Template:Mvar has the same multiplicity in Template:Mvar as it has in Template:Mvar.

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 $\frac{60}{5}=12$ have only 1 as a common factor.

On the contrary, 6 is a divisor but not a unitary divisor of 60, as 6 and $\frac{60}{6}=10$ 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: ${\sigma }^{\ast }(n)$. The sum of the Template:Mvar-th powers of the unitary divisors is denoted by ${\sigma }_k^{\ast }(n)$:

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

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

The number 1 is a unitary divisor of every natural number.

The number of unitary divisors of a number Template:Mvar is Template:Math, where Template:Mvar is the number of distinct prime factors of Template:Mvar. This is because each integer Template:Math is the product of positive powers $p^{r_p}$ of distinct prime numbers Template:Mvar. Thus every unitary divisor of Template:Mvar is the product, over a given subset Template:Mvar of the prime divisors Template:Math} of Template:Mvar, of the prime powers $p^{r_p}$ for Template:Math. If there are Template:Mvar prime factors, then there are exactly Template:Math subsets Template:Mvar, and the statement follows.

The sum of the unitary divisors of Template:Mvar is odd if Template:Mvar is a power of 2 (including 1), and even otherwise.

Both the count and the sum of the unitary divisors of Template:Mvar are multiplicative functions of Template:Mvar that are not completely multiplicative. The Dirichlet generating function is

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

Every divisor of Template:Mvar is unitary if and only if Template:Mvar is square-free.

The set of all unitary divisors of Template:Mvar 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 Template:Mvar forms a Boolean ring, where the addition and multiplication are given by

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

where $(a,b)$ denotes the greatest common divisor of Template:Mvar and Template:Mvar.

Odd unitary divisors

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

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

It is also multiplicative, with Dirichlet generating function

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

Bi-unitary divisors

A divisor Template:Mvar of Template:Mvar is a bi-unitary divisor if the greatest common unitary divisor of Template:Mvar and Template:Math 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 Template:Mvar is a multiplicative function of Template:Mvar with average order $A\log x$ where

$A=\prod _p\left(1-\frac{p-1}{p^2(p+1)}\right)=0.8073308216\cdots .$

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.

OEIS sequences

• Template:OEIS2C is ${\sigma }_0^{\ast }(n)$
• Template:OEIS2C is ${\sigma }_1^{\ast }(n)$
• Template:OEIS2C to Template:OEIS2C are ${\sigma }_2^{\ast }(n)$ to ${\sigma }_8^{\ast }(n)$
• Template:OEIS2C is $2^{\omega }(n)$, the number of unitary divisors
• Template:OEIS2C is ${\sigma }_0^{(o)\ast }(n)$
• Template:OEIS2C is ${\sigma }_1^{(o)\ast }(n)$
• Template:OEIS2C is $\sum _{i=1}^n{\sigma }_1^{\ast }(i)$
• Template:OEIS2C is the constant Template:Mvar

References

• Template:Cite book Section B3.
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• Template:Cite arXiv Section 4.2
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External links

• Template:MathWorld
• Mathoverflow | Boolean ring of unitary divisors

Template:Divisor classes