If a divisor of (that is, ) satisfies , then is a proper divisor of . In the realm of real positive integers, it is usually considered sufficient to list the positive divisors. For example, the proper divisors of 42 are 1, 2, 3, 6, 7, 14, 21.

By restricting the sum of divisors to proper divisors, some will be less than this sum (deficient numbers, including prime numbers), some will be equal (perfect numbers) and some will be greater (abundant numbers). The term restricted divisor is sometimes used to further distinguish divisors in the range (and sometimes it used to mean the same thing as proper divisor). Thus, in our example, the list would be shortened to 2, 3, 6, 7, 14, 21.