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 b / a are coprime, having no common factor other than 1. 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 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 60 5 = 12 {\displaystyle {\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 60 6 = 10 {\displaystyle {\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: σ ∗ ( n ) {\displaystyle \sigma ^{*}(n)} . The sum of the k-th powers of the unitary divisors is denoted by σ k ∗ ( n ) {\displaystyle \sigma _{k}^{*}(n)} :
σ k ∗ ( n ) = ∑ d ∣ n gcd ( d , n / d ) = 1 d k . {\displaystyle \sigma _{k}^{*}(n)=\sum _{d\,\mid \,n \atop \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 n is 2k, 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 {\displaystyle 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 {p} of N, of the prime powers p r p {\displaystyle p^{r_{p}}} for p ∈ S. If there are k prime factors, then there are exactly 2k 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
ζ ( s ) ζ ( s − k ) ζ ( 2 s − k ) = ∑ n ≥ 1 σ k ∗ ( n ) n s . {\displaystyle {\frac {\zeta (s)\zeta (s-k)}{\zeta (2s-k)}}=\sum _{n\geq 1}{\frac {\sigma _{k}^{*}(n)}{n^{s}}}.}
Every divisor of n is unitary if and only if n is square-free. The set of all unitary divisors of n 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 n forms a Boolean ring, where the addition and multiplication are given by
a ⊕ b = a b ( a , b ) 2 , a ⊙ b = ( a , b ) {\displaystyle a\oplus b={\frac {ab}{(a,b)^{2}}},\qquad a\odot b=(a,b)}
where ( a , b ) {\displaystyle (a,b)} denotes the greatest common divisor of a and b.
Odd unitary divisors The sum of the k-th powers of the odd unitary divisors is
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