Divisor -- from Wolfram MathWorld

A divisor, also called a factor, of a number is a number which divides (written ). For integers, only positive divisors are usually considered, though obviously the negative of any positive divisor is itself a divisor. A list of (positive) divisors of a given integer may be returned by the Wolfram Language function Divisors[n].

Sums and products are commonly taken over only some subset of values that are the divisors of a given number. Such a sum would then be denoted, for example,

(1)

Such sums are implemented in the Wolfram Language as DivisorSum[n, form, cond].

The following tables lists the divisors of the first few positive integers (OEIS A027750).

	divisors
1	1
2	1, 2
3	1, 3
4	1, 2, 4
5	1, 5
6	1, 2, 3, 6
7	1, 7
8	1, 2, 4, 8
9	1, 3, 9
10	1, 2, 5, 10
11	1, 11
12	1, 2, 3, 4, 6, 12
13	1, 13
14	1, 2, 7, 14
15	1, 3, 5, 15

The total number of divisors for a given number (variously written , , or ) can be found as follows. Write a number in terms of its prime factorization

(2)

For any divisor of , where

(3)

(4)

Now, , so there are possible values. Similarly, for , there are possible values, so the total number of divisors of is given by

(5)

The product of divisors can be found by writing the number in terms of all possible products

$n={d^((1))d^('(1)); |; d^((nu))d^('(nu)),$

(6)

			(7)
			(8)
			(9)

and

(10)

The geometric mean of divisors is

			(11)
			(12)
			(13)

The arithmetic mean is

(14)

The harmonic mean is

(15)

But , so and

			(16)
			(17)
			(18)

and we have

(19)

(20)

Given three integers chosen at random, the probability that no common factor will divide them all is

(21)

where is Apéry's constant.

The smallest numbers having exactly 0, 1, 2, ... divisors (other than 1) are 1, 2, 4, 6, 16, 12, 64, 24, 36, ... (OEIS A005179; Minin 1883-84; Grost 1968; Roberts 1992, p. 86; Dickson 2005, pp. 51-52). Fontené (1902) and Chalde (1903) showed that if is the prime factorization of the least number with a given number of divisors, then (1) is prime, (2) is prime except for the number which has 8 divisors (Dickson 2005, p. 52).