Function (mathematics)
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Number theory
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Types of functions
Arithmetic function
In number theory, an arithmetic, arithmetical, or number-theoretic function is a real or complex valued function ƒ(n) defined on the set of natural numbers (i.e. positive integers) that "expresses som...
Integer sequence
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Multiplicative function
Integer sequence
In mathematics, an integer sequence is a sequence (i.e., an ordered list) of integers.An integer sequence may be specified explicitly by giving a formula for its nth term, or implicitly by giving a re...
Multiplicative function
In number theory, a multiplicative function is an arithmetic function f(n) of the positive integer n with the property that f(1) = 1 and whenevera and b are coprime, thenAn arithmetic function f(n) is...
Crank of a partition
Arithmetic mean
In mathematics and statistics, the arithmetic mean (/ˌærɪθˈmɛtɪk ˈmiːn/, stress on third syllable of "arithmetic"), or simply the mean or average when the context is clear, is the sum of a collection ...
Von Mangoldt function
In mathematics, the von Mangoldt function is an arithmetic function named after German mathematician Hans von Mangoldt. It is an example of an important arithmetic function that is neither multiplicat...
Von Mangoldt function - Wikipedia
Dirichlet convolution
In mathematics, the Dirichlet convolution is a binary operation defined for arithmetic functions; it is important in number theory. It was developed by Peter Gustav Lejeune Dirichlet, a German mathema...
Dirichlet convolution - Wikipedia
Recurrence relation
In mathematics, a recurrence relation is an equation that recursively defines a sequence or multidimensional array of values, once one or more initial terms are given: each further term of the sequenc...
Geometric mean
In mathematics, the geometric mean is a type of mean or average, which indicates the central tendency or typical value of a set of numbers by using the product of their values (as opposed to the arith...
Geometric mean - Wikipedia
Normal order of an arithmetic function
In number theory, a normal order of an arithmetic function is some simpler or better-understood function which "usually" takes the same or closely approximate values.Let ƒ be a function on the na...
Gauss circle problem
In mathematics, the Gauss circle problem is the problem of determining how many integer lattice points there are in a circle centred at the origin and with radius r. The first progress on a solution ...
List of OEIS sequences
Liouville function
The Liouville function, denoted by λ(n) and named after Joseph Liouville, is an important function in number theory.If n is a positive integer, then λ(n) is defined as:where Ω(n) is the number of...
Factorial
In mathematics, the factorial of a non-negative integer n, denoted by n!, is the product of all positive integers less than or equal to n. For example,The value of 0! is 1, according to the convention...
Factorial - Wikipedia
Euclid number
In mathematics, Euclid numbers are integers of the form En = pn# + 1, where pn# is the nth primorial, i.e. the product of the first n primes. They are named after the ancient Greek mathematician Eucli...
Aliquot sequence
In mathematics, an aliquot sequence is a recursive sequence in which each term is the sum of the proper divisors of the previous term. The aliquot sequence starting with a positive integer k can be de...
Divisibility sequence
In mathematics, a divisibility sequence is an integer sequence such that for all natural numbers m, n,i.e., whenever one index is a multiple of another one, then the corresponding term also...
Fermat number
In mathematics, a Fermat number, named after Pierre de Fermat who first studied them, is a positive integer of the formwhere n is a nonnegative integer. The first few Fermat numbers are:If 2 + 1 is pr...
Fermat number - Wikipedia
Cullen number
In mathematics, a Cullen number is a natural number of the form (written ). Cullen numbers were first studied by Fr. James Cullen in 1905. Cullen numbers are special cases of Proth numbers.
In 19...
Euler's totient function
In number theory, Euler's totient or phi function, φ(n), is an arithmetic function that counts the totatives of n, that is, the positive integers less than or equal to n that are relatively prime to n...
Euler's totient function - Wikipedia
Mertens function
In number theory, the Mertens function is defined for all positive integers n aswhere μ(k) is the Möbius function. The function is named in honour of Franz Mertens.Less formally, M(n) is the count of...
Divisor summatory function
In number theory, the divisor summatory function is a function that is a sum over the divisor function. It frequently occurs in the study of the asymptotic behaviour of the Riemann zeta function. The ...
Divisor summatory function - Wikipedia
Möbius inversion formula
In mathematics, the classic Möbius inversion formula was introduced into number theory during the 19th century by August Ferdinand Möbius. Other Möbius inversion formulas are obtained when differen...
Kronecker's sigma function
Kronecker's sigma function - Wikipedia
Composite number
A composite number is a positive integer that has at least one positive divisor other than one or the number itself. In other words, a composite number is any integer greater than one that is not a pr...
Extremal orders of an arithmetic function
Divisor function
In mathematics, and specifically in number theory, a divisor function is an arithmetic function related to the divisors of an integer. When referred to as the divisor function, it counts the number of...
Average order of an arithmetic function
In number theory, an average order of an arithmetic function is some simpler or better-understood function which takes the same values "on average".Let f be an arithmetic function. We say that an ave...
Cauchy's functional equation
Cauchy's functional equation is the functional equationSolutions to this are called additive functions.Over the rational numbers, it can be shown using elementary algebra that there is a single family...
Incidence algebra
In order theory, a field of mathematics, an incidence algebra is an associative algebra, defined for every locally finite partially ordered setand commutative ring with unity.
A locally finite pos...
Incidence algebra - Wikipedia
Dedekind number
The free distributive lattices of monotonic Boolean functions on 0, 1, 2, and 3 arguments, with 2, 3, 6, and 20 elements respectively In mathematics, the Dedekind numbers are a rapidly growing sequ...
Dedekind number - Wikipedia