Unitary divisor


In mathematics, a natural number is a unitary divisor of a number if is a divisor of and if and are coprime, having no common factor other than 1. Equivalently, a divisor of is a unitary divisor if and only if every prime factor of has the same multiplicity in as it has in.
The concept of a unitary divisor originates from R. Vaidyanathaswamy, who used the term block divisor.

Example

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.

Sum of unitary divisors

The sum-of-unitary-divisors function is denoted by the lowercase Greek letter sigma thus:. The sum of the -th powers of the unitary divisors is denoted by :
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 is, where is the number of distinct prime factors of.
This is because each integer is the product of positive powers of distinct prime numbers. Thus every unitary divisor of is the product, over a given subset of the prime divisors of, of the prime powers for. If there are prime factors, then there are exactly subsets, and the statement follows.
The sum of the unitary divisors of is odd if is a power of 2, and even otherwise.
Both the count and the sum of the unitary divisors of are multiplicative functions of that are not completely multiplicative. The Dirichlet generating function is
Every divisor of is unitary if and only if is square-free.
The set of all unitary divisors of 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 forms a Boolean ring, where the addition and multiplication are given by
where denotes the greatest common divisor of and.

Odd unitary divisors

The sum of the k-th powers of the odd unitary divisors is
It is also multiplicative, with Dirichlet generating function

Bi-unitary divisors

A divisor of is a bi-unitary divisor if the greatest common unitary divisor of and is 1. This concept originates from D. Suryanarayana. .
The number of bi-unitary divisors of is a multiplicative function of with average order where
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

  • is
  • is
  • to are to
  • is, the number of unitary divisors
  • is
  • is
  • is
  • is the constant