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Unitary divisor

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Unitary divisor

In mathematics, a natural number a is a unitary divisor of a number b if a is a divisor of b and if a and b a are coprime, having no common factor other than 1. Thus, 5 is a unitary divisor of 60, because 5 and 60 5 = 12 have only 1 as a common factor, while 6 is a divisor but not a unitary divisor of 60, as 6 and 60 6 = 10 have a common factor other than 1, namely 2. 1 is a unitary divisor of every natural number.

Contents

Equivalently, a given 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):

σ k ∗ ( n ) = ∑ d ∣ n gcd ( d , n / d ) = 1 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 2k, where k is the number of distinct prime factors of n. 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

ζ ( s ) ζ ( s − k ) ζ ( 2 s − k ) = ∑ n ≥ 1 σ k ∗ ( n ) n s .

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

σ k ( o ) ∗ ( n ) = ∑ d ∣ n d ≡ 1 ( mod 2 ) gcd ( d , n / d ) = 1 d k .

It is also multiplicative, with Dirichlet generating function

ζ ( s ) ζ ( s − k ) ( 1 − 2 k − s ) ζ ( 2 s − k ) ( 1 − 2 k − 2 s ) = ∑ n ≥ 1 σ k ( o ) ∗ ( n ) n s .

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 where

A = ∏ p ( 1 − p − 1 p 2 ( p + 1 ) )   .

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.

References

Unitary divisor Wikipedia
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