In
mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics ...
, and specifically in
number theory
Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic function, integer-valued functions. German mathematician Carl Friedrich Gauss (1777â ...
, a divisor function is an
arithmetic function
In number theory, an arithmetic, arithmetical, or number-theoretic function is for most authors any function ''f''(''n'') whose domain is the positive integers and whose range is a subset of the complex numbers. Hardy & Wright include in their d ...
related to 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 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 ...
. When referred to as ''the'' divisor function, it counts the ''number of divisors of an integer'' (including 1 and the number itself). It appears in a number of remarkable identities, including relationships on the
Riemann zeta function
The Riemann zeta function or Euler–Riemann zeta function, denoted by the Greek letter (zeta), is a mathematical function of a complex variable defined as \zeta(s) = \sum_^\infty \frac = \frac + \frac + \frac + \cdots for \operatorname(s) > ...
and the
Eisenstein series
Eisenstein series, named after German mathematician Gotthold Eisenstein, are particular modular forms with infinite series expansions that may be written down directly. Originally defined for the modular group, Eisenstein series can be generaliz ...
of
modular form
In mathematics, a modular form is a (complex) analytic function on the upper half-plane satisfying a certain kind of functional equation with respect to the Group action (mathematics), group action of the modular group, and also satisfying a grow ...
s. Divisor functions were studied by
Ramanujan, who gave a number of important
congruences
In abstract algebra, a congruence relation (or simply congruence) is an equivalence relation on an algebraic structure (such as a group, ring, or vector space) that is compatible with the structure in the sense that algebraic operations done wi ...
and
identities; these are treated separately in the article
Ramanujan's sum
In number theory, Ramanujan's sum, usually denoted ''cq''(''n''), is a function of two positive integer variables ''q'' and ''n'' defined by the formula
: c_q(n) = \sum_ e^,
where (''a'', ''q'') = 1 means that ''a'' only takes on values coprime ...
.
A related function is the
divisor summatory function
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 ...
, which, as the name implies, is a sum over the divisor function.
Definition
The sum of positive divisors function σ
''z''(''n''), for a real or complex number ''z'', is defined as the
sum of the ''z''th
powers of the positive
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 ''n''. It can be expressed in
sigma notation
In mathematics, summation is the addition of a sequence of any kind of numbers, called ''addends'' or ''summands''; the result is their ''sum'' or ''total''. Beside numbers, other types of values can be summed as well: functions, vectors, matr ...
as
:
where
is shorthand for "''d''
divides
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 ...
''n''".
The notations ''d''(''n''), ν(''n'') and τ(''n'') (for the German ''Teiler'' = divisors) are also used to denote σ
0(''n''), or the number-of-divisors function
(). When ''z'' is 1, the function is called the sigma function or sum-of-divisors function,
and the subscript is often omitted, so σ(''n'') is the same as σ
1(''n'') ().
The
aliquot sum
In number theory, the aliquot sum ''s''(''n'') of a positive integer ''n'' is the sum of all proper divisors of ''n'', that is, all divisors of ''n'' other than ''n'' itself.
That is,
:s(n)=\sum\nolimits_d.
It can be used to characterize the prim ...
''s''(''n'') of ''n'' is the sum of the
proper 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 (that is, the divisors excluding ''n'' itself, ), and equals σ
1(''n'') − ''n''; the
aliquot sequence
In mathematics, an aliquot sequence is a sequence of positive integers in which each term is the sum of the proper divisors of the previous term. If the sequence reaches the number 1, it ends, since the sum of the proper divisors of 1 is 0.
Defi ...
of ''n'' is formed by repeatedly applying the aliquot sum function.
Example
For example, σ
0(12) is the number of the divisors of 12:
:
while σ
1(12) is the sum of all the divisors:
:
and the aliquot sum s(12) of proper divisors is:
:
σ
-1(''n'') is sometimes called the
abundancy index
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 ...
of ''n'', and we have:
:
Table of values
The cases ''x'' = 2 to 5 are listed in − , ''x'' = 6 to 24 are listed in − .
Properties
Formulas at prime powers
For 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 ...
''p'',
:
because by definition, the factors of a prime number are 1 and itself. Also, where ''p
n''# denotes the
primorial
In mathematics, and more particularly in number theory, primorial, denoted by "#", is a function from natural numbers to natural numbers similar to the factorial function, but rather than successively multiplying positive integers, the function ...
,
:
since ''n'' prime factors allow a sequence of binary selection (
or 1) from ''n'' terms for each proper divisor formed. However, these are not in general the smallest numbers whose number of divisors is a
power of two
A power of two is a number of the form where is an integer, that is, the result of exponentiation with number two as the base and integer as the exponent.
In a context where only integers are considered, is restricted to non-negative ...
; instead, the smallest such number may be obtained by multiplying together the first ''n''
Fermi–Dirac prime
In number theory, a Fermi–Dirac prime is a prime power whose exponent is a power of two. These numbers are named from an analogy to Fermi–Dirac statistics in physics based on the fact that each integer has a unique representation as a product ...
s, prime powers whose exponent is a power of two.
Clearly,
for all
, and
for all
,
.
The divisor function is
multiplicative (since each divisor ''c'' of the product ''mn'' with
distinctively correspond to a divisor ''a'' of ''m'' and a divisor ''b'' of ''n''), but not
completely multiplicative In number theory, functions of positive integers which respect products are important and are called completely multiplicative functions or totally multiplicative functions. A weaker condition is also important, respecting only products of coprime ...
:
:
The consequence of this is that, if we write
:
where ''r'' = ''ω''(''n'') is the
number of distinct prime factors of ''n'', ''p
i'' is the ''i''th prime factor, and ''a
i'' is the maximum power of ''p
i'' by which ''n'' is
divisible
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 ...
, then we have:
:
which, when ''x'' ≠ 0, is equivalent to the useful formula:
:
When ''x'' = 0, ''d''(''n'') is:
:
This result can be directly deduced from the fact that all divisors of
are uniquely determined by the distinct tuples
of integers with
(i.e.
independent choices for each
).
For example, if ''n'' is 24, there are two prime factors (''p
1'' is 2; ''p
2'' is 3); noting that 24 is the product of 2
3×3
1, ''a''
1 is 3 and ''a''
2 is 1. Thus we can calculate
as so:
:
The eight divisors counted by this formula are 1, 2, 4, 8, 3, 6, 12, and 24.
Other properties and identities
Euler
Leonhard Euler ( , ; 15 April 170718 September 1783) was a Swiss mathematician, physicist, astronomer, geographer, logician and engineer who founded the studies of graph theory and topology and made pioneering and influential discoveries in ma ...
proved the remarkable recurrence:
:
where
if it occurs and
for
, and
are consecutive pairs of generalized
pentagonal numbers
A pentagonal number is a figurate number that extends the concept of triangular and square numbers to the pentagon, but, unlike the first two, the patterns involved in the construction of pentagonal numbers are not rotationally symmetrical. The ...
(, starting at offset 1). Indeed, Euler proved this by logarithmic differentiation of the identity in his
Pentagonal number theorem
In mathematics, the pentagonal number theorem, originally due to Euler, relates the product and series representations of the Euler function. It states that
:\prod_^\left(1-x^\right)=\sum_^\left(-1\right)^x^=1+\sum_^\infty(-1)^k\left(x^+x^\right ...
.
For a non-square integer, ''n'', every divisor, ''d'', of ''n'' is paired with divisor ''n''/''d'' of ''n'' and
is even; for a square integer, one divisor (namely
) is not paired with a distinct divisor and
is odd. Similarly, the number
is odd if and only if ''n'' is a square or twice a square.
We also note ''s''(''n'') = ''σ''(''n'') − ''n''. Here ''s''(''n'') denotes the sum of the ''proper'' divisors of ''n'', that is, the divisors of ''n'' excluding ''n'' itself. This function is used to recognize
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 ...
s, which are the ''n'' such that ''s''(''n'') = ''n''. If ''s''(''n'') > ''n'', then ''n'' is 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 ...
, and if ''s''(''n'') < ''n'', then ''n'' is 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 ...
.
If is a power of 2,
, then
and
, which makes ''n''
almost-perfect.
As an example, for two primes