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.

The integer 5 is a unitary divisor of 60, because 5 and ${\displaystyle {\frac {60}{5}}=12}$ have only 1 as a common factor.

On the contrary, 6 is a divisor but not a unitary divisor of 60, as 6 and ${\displaystyle {\frac {60}{6}}=10}$ have a common factor other than 1, namely 2.

The sum-of-unitary-divisors function is denoted by the lowercase Greek letter sigma thus: ${\displaystyle \sigma ^{*}(n)}$. The sum of the k-th powers of the unitary divisors is denoted by ${\displaystyle \sigma _{k}^{*}(n)}$:

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.

Number 1 is a unitary divisor of every natural number.

The number of unitary divisors of a number n is 2^k, where k is the number of distinct prime factors of n. This is because each integer N > 1 is the product of positive powers ${\displaystyle p^{r_{p}}}$ 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 , of the prime powers ${\displaystyle p^{r_{p}}}$ for p ∈ S. If there are k prime factors, then there are exactly 2^k subsets S, and the statement follows.