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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
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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):
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
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
It is also multiplicative, with Dirichlet generating function
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 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.