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. In the
formulas below, 
is a positive integer and all divisors are positive.
A list of these divisors 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)
|
so
 |
(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
 |
(6)
|
so
 |  | ![[d^((1))...d^((nu))][d^('(1))...d^('(nu))]](/images/equations/Divisor/Inline23.svg) |
(7)
|
 |  |  |
(8)
|
 |  |  |
(9)
|
and
 |
(10)
|
The geometric mean of divisors is
 |  |  |
(11)
|
 |  | ![[n^(nu(n)/2)]^(1/nu(n))](/images/equations/Divisor/Inline35.svg) |
(12)
|
 |  |  |
(13)
|
The arithmetic mean is
 |
(14)
|
The harmonic mean is
 |
(15)
|
But 
, so 
and
 |  |  |
(16)
|
 |  |  |
(17)
|
 |  |  |
(18)
|
and we have
 |
(19)
|
 |
(20)
|
For three integers chosen independently and uniformly from 1, 2, ..., 
,
the probability that their greatest
common divisor is 1 tends, as 
, to
![[zeta(3)]^(-1) approx 1.20206^(-1) approx 0.831907,](/images/equations/Divisor/NumberedEquation12.svg) |
(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, with
the primes in increasing order and the exponents
in decreasing order, then (1) 
is prime, and (2), for 
, 
is prime except for the number 
, which has 8 divisors (Dickson 2005, p. 52).
Let 
be the number of elements in the
greatest subset of ![[1,n]](/images/equations/Divisor/Inline59.svg)
such that none of its elements are divisible by two others. For 
sufficiently large,
 |
(22)
|
(Le Lionnais 1983, Lebensold 1976/1977).
Coprimality can also be imposed when matching the divisors of an integer to an arithmetic progression. Let 
be a positive integer,
let 
be its positive divisors, let 
be their number, and let 
for integers 
and 
. Alekseyev (2026) proved that there is a bijection 
with 
for every 
iff, for every prime 
dividing 
,
 |
(23)
|
where 
is the exponent
of 
in the prime
factorization of 
.
Alekseyev (2026) reports AI assistance with developing and checking the proof.
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