Table of divisors
The tables below list all of the divisors of the numbers 1 to 1000.
A divisor of an integer n is an integer m, for which n/m is again an integer (which is necessarily also a divisor of n). For example, 3 is a divisor of 21, since 21/7 = 3 (and therefore 7 is also a divisor of 21).
If m is a divisor of n, then so is −m. The tables below only list positive divisors.
Key to the tables
- d(n) is the number of positive divisors of n, including 1 and n itself
- σ(n) is the sum of the positive divisors of n, including 1 and n itself
- s(n) is the sum of the proper divisors of n, including 1 but not n itself; that is, s(n) = σ(n) − n
- a deficient number is greater than the sum of its proper divisors; that is, s(n) < n
- a perfect number equals the sum of its proper divisors; that is, s(n) = n
- an abundant number is lesser than the sum of its proper divisors; that is, s(n) > n
- a highly abundant number has a sum of positive divisors that is greater than any lesser number; that is, σ(n) > σ(m) for every positive integer m < n. Counterintuitively, the first seven highly abundant numbers are not abundant numbers.
- a prime number has only 1 and itself as divisors; that is, d(n) = 2
- a composite number has more than just 1 and itself as divisors; that is, d(n) > 2
- a highly composite number has a number of positive divisors that is greater than any lesser number; that is, d(n) > d(m) for every positive integer m < n. Counterintuitively, the first two highly composite numbers are not composite numbers.
- a superior highly composite number has a ratio between its number of divisors and itself raised to some positive power that equals or is greater than any other number; that is, there exists some ε such that for every other positive integer m
- a primitive abundant number is an abundant number whose proper divisors are all deficient numbers
- a weird number is an abundant number that is not semiperfect; that is, no subset of the proper divisors of n sum to n
