The tables below list all of the

divisor In mathematics, a divisor of an integer n, also called a factor of n, is an integer m that may be multiplied by some integer to produce n. In this case, one also says that n is a multiple of m. An integer n is divisible or evenly divisible by ...

s of the numbers 1 to 1000. A divisor of an

integer An integer is the number zero (), a positive natural number (, , , etc.) or a negative integer with a minus sign (−1, −2, −3, etc.). The negative numbers are the additive inverses of the corresponding positive numbers. In the language ...

''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 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 In mathematics, a divisor of an integer n, also called a factor of n, is an integer m that may be multiplied by some integer to produce n. In this case, one also says that n is a multiple of m. An integer n is divisible or evenly divisible by ...

of ''n'', including 1, but not ''n'' itself; that is, ''s''(''n'') = σ(''n'') − ''n'' *a

deficient number In number theory, a deficient number or defective number is a number ''n'' for which the sum of divisors of ''n'' is less than 2''n''. Equivalently, it is a number for which the sum of proper divisors (or aliquot sum) is less than ''n''. For e ...

is greater than the sum of its proper divisors; that is, ''s''(''n'') < ''n'' *a

perfect number In number theory, a perfect number is a positive integer that is equal to the sum of its positive divisors, excluding the number itself. For instance, 6 has divisors 1, 2 and 3 (excluding itself), and 1 + 2 + 3 = 6, so 6 is a perfect number. T ...

equals the sum of its proper divisors; that is, ''s''(''n'') = ''n'' *an

abundant number In number theory, an abundant number or excessive number is a number for which the sum of its proper divisors is greater than the number. The integer 12 is the first abundant number. Its proper divisors are 1, 2, 3, 4 and 6 for a total of 16. The ...

is lesser than the sum of its proper divisors; that is, ''s''(''n'') > ''n'' *a

highly abundant number In mathematics, a highly abundant number is a natural number with the property that the sum of its divisors (including itself) is greater than the sum of the divisors of any smaller natural number. Highly abundant numbers and several similar cla ...

has a sum of positive divisors greater than any lesser number's sum of positive divisors; that is, ''s(n) > s(m) for every positive integer m < n''. Counterintuitively, the first seven ''highly abundant'' numbers are not ''abundant'' numbers. *a

prime number A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways ...

has only 1 and itself as divisors; that is, ''d''(''n'') = 2. Prime numbers are always deficient as ''s''(''n'')=1 *a

composite number A composite number is a positive integer that can be formed by multiplying two smaller positive integers. Equivalently, it is a positive integer that has at least one divisor other than 1 and itself. Every positive integer is composite, prime, ...

has more than just 1 and itself as divisors; that is, ''d''(''n'') > 2 *a

highly composite number __FORCETOC__ A highly composite number is a positive integer with more divisors than any smaller positive integer has. The related concept of largely composite number refers to a positive integer which has at least as many divisors as any smaller ...

has more divisors 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 In mathematics, a superior highly composite number is a natural number which has the highest ratio of its number of divisors to ''some'' positive power of itself than any other number. It is a stronger restriction than that of a highly composite ...

has more divisors than any other number scaled relative to some positive power of the number itself; that is, ''there exists some ε such that''

\frac>\frac

''for every other positive integer m.'' Superior highly composite numbers are always highly composite numbers *a

weird number In number theory, a weird number is a natural number that is abundant but not semiperfect. In other words, the sum of the proper divisors (divisors including 1 but not itself) of the number is greater than the number, but no subset of those diviso ...

is an abundant number that is not

semiperfect In number theory, a semiperfect number or pseudoperfect number is a natural number ''n'' that is equal to the sum of all or some of its proper divisors. A semiperfect number that is equal to the sum of all its proper divisors is a perfect number. ...

; that is, ''no subset of the proper divisors of n sum to n''

1 to 100

101 to 200

201 to 300

301 to 400

401 to 500

501 to 600

601 to 700

701 to 800

801 to 900

901 to 1000

External links

* {{OEIS el, 1=A027750, 2=Triangle read by rows in which row n lists the divisors of n Divisor function Elementary number theory

Divisors In mathematics, a divisor of an integer n, also called a factor of n, is an integer m that may be multiplied by some integer to produce n. In this case, one also says that n is a multiple of m. An integer n is divisible or evenly divisible by ...

Numbers