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mathematics, an integer-valued function is a function whose values are integers. In other words, it is a function that assigns an integer to each member
Integer-valued_function
Nearest integers from a number
and ceiling functions In mathematics, the floor function is the function that takes a real number x as input and returns the greatest integer less than
Floor_and_ceiling_functions
Topics referred to by the same term
Integer function may refer to: Integer-valued function, an integer function Floor function, sometimes referred as the integer function, INT Arithmetic
Integer_function
Number in {..., –2, –1, 0, 1, 2, ...}
a positive integer Complex integer Hyperinteger Integer complexity Integer lattice Integer part Integer sequence Integer-valued function Mathematical
Integer
Generalized mathematical function
a multivalued function, multiple-valued function, many-valued function, or multifunction, is a function that has two or more values in its range for
Multivalued_function
Function returning one of only two values
a Boolean function is a k-ary integer-valued function giving the correlation between a certain set of changes in the inputs and the function output. For
Boolean_function
Family of solutions to related differential equations
is an integer or a half-integer. When α {\displaystyle \alpha } is an integer, the resulting Bessel functions are often called cylinder functions or cylindrical
Bessel_function
Replacing a number with a simpler value
especially when dividing two numbers in integer or fixed-point arithmetic; when computing mathematical functions such as square roots, logarithms, and sines;
Rounding
Constants of the mathematical zeta function
It also includes derivatives and some series composed of the zeta function at integer arguments. The same equation in s {\displaystyle s} above also holds
Particular values of the Riemann zeta function
Particular_values_of_the_Riemann_zeta_function
This is a list of notable integer sequences with links to their entries in the On-Line Encyclopedia of Integer Sequences. OEIS core sequences Index to
List_of_integer_sequences
Degree of differentiability of a function or map
discontinuities. The smoothness, or differentiability class, is an integer k such that a function has all derivatives up to order k, and such that all of these
Smoothness
Function in mathematical number theory
the Carmichael function λ(n) of a positive integer n is the smallest positive integer m such that am≡1(modn) holds for every integer a coprime to n.
Carmichael_function
Mathematical constants
gamma function is an important special function in mathematics. Its particular values can be expressed in closed form for integer, half-integer, and some
Particular values of the gamma function
Particular_values_of_the_gamma_function
Function that is continuous everywhere but differentiable nowhere
mathematics, the Weierstrass function, named after its discoverer, Karl Weierstrass, is an example of a real-valued function that is continuous everywhere
Weierstrass function (nowhere-differentiable function)
Weierstrass_function_(nowhere-differentiable_function)
Extension of the factorial function
OEIS. The values presented here are truncated rather than rounded.) The complex-valued gamma function is undefined for non-positive integers, but in these
Gamma_function
Function with a multiplicative scaling behaviour
degree of homogeneity, or simply the degree. That is, if k is an integer, a function f of n variables is homogeneous of degree k if f ( s x 1 , … , s
Homogeneous_function
Functions such that f(–x) equals f(x) or –f(x)
n is an odd integer. Even functions are those real functions whose graph is self-symmetric with respect to the y-axis, and odd functions are those whose
Even_and_odd_functions
Mapping arbitrary data to fixed-size values
A hash function is any function that can be used to map data of arbitrary size to fixed-size values, though there are some hash functions that support
Hash_function
Arithmetic operation
non-commutative exponentiation. For nonnegative integers n and m, the value of nm is the number of functions from a set of m elements to a set of n elements
Exponentiation
Types of special mathematical functions
the domain C of multi-valued functions by a suitable manifold in C × C called Riemann surface. While this removes multi-valuedness, one has to know the
Incomplete_gamma_function
Association of one output to each input
scalar-valued or vector-valued functions, which share a specific property and form a topological vector space. For example, the real smooth functions with
Function_(mathematics)
Polynomial with integer values for integer inputs
mathematics, an integer-valued polynomial (also known as a numerical polynomial) is a polynomial whose value is an integer for every integer input. Every
Integer-valued_polynomial
simple reflection has length one. The function l is then an integer-valued function of W; it is a length function of W. It follows immediately from the
Length of a Weyl group element
Length_of_a_Weyl_group_element
Computational operation
arguments, a and m, being integers, many systems now allow other types of numeric operands. When they are both integers, the range of values for a mod m {\displaystyle
Modulo
Open problem on 3x+1 and x/2 functions
positive integer: If the number is even, divide it by two. If the number is odd, triple it and add one. In modular arithmetic notation, define the function f
Collatz_conjecture
Method to solve optimization problems
is defined by a linear inequality. Its objective function is a real-valued affine (linear) function defined on this polytope. A linear programming algorithm
Linear_programming
Number of partitions of an integer
partition function p(n) represents the number of possible partitions of a non-negative integer n. For instance, p(4) = 5 because the integer 4 has the
Partition function (number theory)
Partition_function_(number_theory)
Function representing the number of primes less than or equal to a given number
of integers less than or equal to x which are divisible by no pi is ⌊x⌋−∑i⌊xpi⌋+∑i<j⌊xpipj⌋−∑i<j<k⌊xpipjpk⌋+⋯ (where ⌊x⌋ denotes the floor function). This
Prime-counting_function
Analytic function in mathematics
Riemann sphere the zeta function has an essential singularity. For sums involving the zeta function at integer and half-integer values, see rational zeta series
Riemann_zeta_function
Arithmetic function related to the divisors of an integer
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
Divisor_function
Data types supported by the C programming language
and false. _Bool functions similarly to a normal integer type, with one exception: any conversion to a _Bool gives 0 (false) if the value equals 0; otherwise
C_data_types
Inverse functions of sin, cos, tan, etc.
-\arctan(x)}{\pi }}\right)\,.} The function rni {\displaystyle \operatorname {rni} } rounds to the nearest integer. For angles near 0 and π, arccosine
Inverse trigonometric functions
Inverse_trigonometric_functions
Concept in probability theory and statistics
theory and statistics, the moment generating function of a real-valued random variable is a generating function that provides an alternative specification
Moment_generating_function
Decomposition of an integer as a sum of positive integers
partition of a non-negative integer n, also called an integer partition, is a way of writing n as a sum of positive integers. Two sums that differ only
Integer_partition
Point to which functions converge in analysis
limit of af(x) as x approaches p is aL. If f and g are real-valued (or complex-valued) functions, then taking the limit of an operation on f(x) and g(x) (e
Limit_of_a_function
Whose values lie in an infinite-dimensional vector space
Such functions are applied in most sciences including physics. Set f k ( t ) = t / k 2 {\displaystyle f_{k}(t)=t/k^{2}} for every positive integer k {\displaystyle
Infinite-dimensional vector function
Infinite-dimensional_vector_function
Datum of integral data type
negative values. Integers are commonly represented in a computer as a group of binary digits (bits). The size of the grouping varies so the set of integer sizes
Integer_(computer_science)
Problem in number theory on equal totients
mathematics, Carmichael's totient function conjecture concerns the multiplicity of values of Euler's totient function φ ( n ) {\displaystyle \varphi (n)}
Carmichael's totient function conjecture
Carmichael's_totient_function_conjecture
Type of function in mathematics
{\displaystyle 0} changes its value by an integer multiple of 2 π i {\displaystyle 2\pi i} . For this reason, a single-valued branch of the logarithm can
Analytic_function
Subfield of number theory
explicitly uses probability to answer questions about the integers and integer-valued functions. One basic idea underlying it is that different prime numbers
Probabilistic_number_theory
Mathematical function whose derivative exists
or complex function of a single variable is differentiable if its derivative exists at each point in its domain. For real-valued functions of a real variable
Differentiable_function
Function on an integer n which is log(p) if n equals p^k and zero otherwise
Mangoldt function, denoted by Λ ( n ) {\displaystyle \Lambda (n)} , is defined as Λ ( n ) = { log p if n = p k for some prime p and integer k ≥ 1
Von_Mangoldt_function
Type of mathematical expression
Schmidt, ISBN 0-534-93219-3 Cahen, Paul-Jean; Chabert, Jean-Luc (1997). Integer-Valued Polynomials. American Mathematical Society. ISBN 978-0-8218-0388-2.
Polynomial
Computer arithmetic error
computer programming, an integer overflow occurs when an arithmetic operation on integers attempts to create a numeric value that is outside of the range
Integer_overflow
Probability that random variable X is less than or equal to x
cumulative distribution function (CDF) of a real-valued random variable X {\displaystyle X} , or just distribution function of X {\displaystyle X} ,
Cumulative distribution function
Cumulative_distribution_function
Type of mathematical function
Dirichlet L-functions may be written as linear combinations of the Hurwitz zeta function at rational values. Fixing an integer k≥1, Dirichlet L-functions for
Dirichlet_L-function
Multivalued function in mathematics
Building on Lambert's work, Leonhard Euler described the W function per se in 1783. For each integer k {\displaystyle k} there is one branch, denoted by W
Lambert_W_function
Function that is holomorphic on the whole complex plane
In complex analysis, an entire function, also called an integral function, is a complex-valued function that is holomorphic on the whole complex plane
Entire_function
Integer
inside the function f, its inverse will yield an inverse image, or preimage, of that subset under the function. Exponentiation to negative integers can be
−1
Conjecture in group theory
{\displaystyle \alpha } in S {\displaystyle S} , then there exists an integer-valued function f {\displaystyle f} such that f ( d ) ≥ | G α | {\displaystyle
Sims_conjecture
Mathematical optimization problem restricted to integers
are restricted to be integers. In many settings the term refers to integer linear programming (ILP), in which the objective function and the constraints
Integer_programming
Function that is discontinuous at rationals and continuous at irrationals
Thomae's function is a real-valued function of a real variable that can be defined as: f ( x ) = { 1 q if x = p q ( x is rational), with p ∈ Z and
Thomae's_function
Special mathematical function defined as sin(x)/x
normalized sinc function are the nonzero integer values of x. The function has also been called the cardinal sine or sine cardinal function The term "sinc"
Sinc_function
Number of integers coprime to and less than n
_{e}(x)} . In number theory, Euler's totient function counts the positive integers up to a given integer n {\displaystyle n} that are relatively prime
Euler's_totient_function
Function whose domain is the positive integers
arithmetical, or number-theoretic function is generally any function whose domain is the set of positive integers and whose range is a subset of the
Arithmetic_function
Functions of an angle
\sin(x+y).} A positive integer appearing as a superscript after the symbol of the function denotes exponentiation, not function composition. For example
Trigonometric_functions
C standard library header file
mathematical functions, which use floating-point numbers, are defined in <math.h> (<cmath> header in C++). The functions that operate on integers, such as
C_mathematical_functions
Logarithm to the base of the mathematical constant e
the values being expressed as integers. The natural logarithm can be defined more generally as the inverse function of the exponential function ex, so
Natural_logarithm
Computer programming language
basic types are integer, float, string, null, table, array, function, generator, class, instance, bool, thread and userdata. An Integer represents a 32
Squirrel (programming language)
Squirrel_(programming_language)
Root of a quadratic polynomial with a unit leading coefficient
theory, quadratic integers are a generalization of the usual integers to quadratic fields. A complex number is called a quadratic integer if it is a root
Quadratic_integer
Number used for counting
are 0 (if included), 1, 2, 3, and so on. The terms positive integers, non-negative integers, whole numbers, and counting numbers are also used. The set
Natural_number
Mathematical functions
inverse function while (sinhx)−1 or sinh(x)−1 means the reciprocal 1/sinhx. Especially inconsistent is the conventional use of positive integer superscripts
Inverse_hyperbolic_functions
Programming language evaluation rules
kind of value that is passed to the function for each parameter (the binding strategy) and whether to evaluate the parameters of a function call, and
Evaluation_strategy
Ordered list of whole numbers
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
Integer_sequence
Logarithm of a complex number
k\right)} for integers k {\displaystyle k} . These logarithms are equally spaced along a vertical line in the complex plane. A complex-valued function log : U
Complex_logarithm
Online database of integer sequences
The On-Line Encyclopedia of Integer Sequences (OEIS) is an online database of integer sequences. It was created and maintained by Neil Sloane while researching
On-Line Encyclopedia of Integer Sequences
On-Line_Encyclopedia_of_Integer_Sequences
Statistical method of dividing data into equal-sized intervals for analysis
a sample of size N by computing a real valued index h. When h is an integer, the h-th smallest of the N values, xh, is the quantile estimate. Otherwise
Quantile
Product of numbers from 1 to n
factorial function to a continuous function of complex numbers, except at the negative integers, the (offset) gamma function. Many other notable functions and
Factorial
Attribute of data
-> Bool denoting functions taking an integer and returning a Boolean. In C, a function is not a first-class data type but function pointers can be manipulated
Data_type
Rational number equal to an integer plus 1/2
{\pi ^{n/2}}{\Gamma ({\frac {n}{2}}+1)}}R^{n}~.} The values of the gamma function on half-integers are rational multiples of the square root of pi: Γ (
Half-integer
Variable that represents an argument to a function
itself, as in this function, which computes the size of a text fragment: TextExtent(WString text, Font font : Integer width, Integer height) Parameter
Parameter (computer programming)
Parameter_(computer_programming)
Finite or infinite ordered list of elements
doubly infinite sequence. A function from ℤ the set of all integers, into a set, for example the sequence of all even integers (..., −4, −2, 0, 2, 4, 6,
Sequence
the power of a positive integer. Constant function: polynomial of degree zero, graph is a horizontal straight line Linear function: First degree polynomial
List of mathematical functions
List_of_mathematical_functions
Mathematical function, inverse of an exponential function
tends to be a multi-valued function. For example, the complex logarithm is the multi-valued inverse of the complex exponential function. Similarly, the discrete
Logarithm
Quantum mechanics principle
exclusion principle states that two or more identical particles with half-integer spins (i.e., fermions) cannot simultaneously occupy the same quantum state
Pauli_exclusion_principle
Counterexample to the converse of the intermediate value theorem
13 function is a mathematical function created by British mathematician John H. Conway as a counterexample to the converse of the intermediate value theorem
Conway's_base_13_function
Property of functions which is weaker than continuity
semi-continuity) is a property of extended real-valued functions that is weaker than continuity. An extended real-valued function f {\displaystyle f} is upper (respectively
Semi-continuity
Number with a real and an imaginary part
numbers are often used to compute certain real-valued improper integrals, by means of complex-valued functions. Several methods exist to do this; see methods
Complex_number
Type of mathematical function
sheaves of locally constant functions on X . {\displaystyle X.} To be more definite, the locally constant integer-valued functions on X {\displaystyle X} form
Locally_constant_function
Non-cryptographic hash function
unsigned integer. The FNV_offset_basis is the 64-bit value: 14695981039346656037 (in hex, 0xcbf29ce484222325). The FNV_prime is the 64-bit value 1099511628211
Fowler–Noll–Vo_hash_function
Number without repeated prime factors
In mathematics, a square-free integer (or squarefree integer) is an integer that is divisible by no square number other than 1. That is, its prime factorization
Square-free_integer
Product of a number by itself
It is easier to compute than the absolute value (no square root), and is smooth as a real-valued function. Because of these two properties, the absolute
Square_(algebra)
C function to format and output text
number of value arguments that the function serializes per the format string. Mismatch between the format specifiers and count and type of values results
Printf
to calculate the mean value in terms of the Riemann zeta function. This is illustrated in the following example. For an integer k ≥ 1 {\displaystyle k\geq
Average order of an arithmetic function
Average_order_of_an_arithmetic_function
Solution of a confluent hypergeometric equation
poles at the non-positive integers. Some values of a and b yield solutions that can be expressed in terms of other known functions. See #Special cases. When
Confluent hypergeometric function
Confluent_hypergeometric_function
Natural number
aliquot numbers: impossible values for the sum of aliquot parts function (A001065))". The On-Line Encyclopedia of Integer Sequences. OEIS Foundation.
300_(number)
represents a boolean as an integer where false is represented as a value of zero and true by a non-zero value. ^c All values evaluate to either true or
Comparison of programming languages (basic instructions)
Comparison_of_programming_languages_(basic_instructions)
Classes of data types
value type) into an Integer object (an object type), or reversing this via "unboxing". Even when function arguments are passed using "call by value"
Value_type_and_reference_type
Quickly growing function
function (which had three non-negative integer arguments), many authors modified it to suit various purposes, so that today "the Ackermann function"
Ackermann_function
Integers have unique prime factorizations
factorization theorem and prime factorization theorem, states that every integer greater than 1 is either prime or can be represented uniquely as a product
Fundamental theorem of arithmetic
Fundamental_theorem_of_arithmetic
Mathematical function, denoted exp(x) or e^x
mathematics, the exponential function is the unique real function which maps zero to one and has a derivative everywhere equal to its value. It is denoted e x
Exponential_function
real-valued functions defined on a topological space, as follows. The Baire class 0 functions are the continuous functions. The Baire class 1 functions are
Baire_function
Point of interest for complex multi-valued functions
point of a multivalued function is a point such that if the function is n {\displaystyle n} -valued (has n {\displaystyle n} values) at that point, all of
Branch_point
On integer partitions from monotonic functions
f {\displaystyle f} be any function from positive integers to non-negative integers that is both non-decreasing (each value in the sequence f ( 1 ) , f
Lambek–Moser_theorem
Inputs for which a function's value is non-zero
In mathematics, the support of a real-valued function f is the subset of the function's domain consisting of those elements that are not mapped to zero
Support_(mathematics)
char* argv[]) { IntegerPair pairArray[] = { {0, 1}, {2, 3}, {4, 5} }; printElements(a, 3); // the anonymous function is given as function for the foreach
Examples of anonymous functions
Examples_of_anonymous_functions
Function used in signal processing
statistics, a window function (also known as an apodization function or tapering function) is a mathematical function that is zero-valued outside of some chosen
Window_function
Technique for selecting hash functions
before picking a new hash function). A survey of fastest known universal and strongly universal hash functions for integers, vectors, and strings is found
Universal_hashing
Mathematical concept
example, consider the real-valued function of a real variable given by f(x) = 5x − 7. One can think of f as the function which multiplies its input by
Inverse_function
Numbers significantly larger than those used regularly
conventional function notation, can also be used with the same functions. Fast-growing hierarchy These notations are essentially functions of integer variables
Large_numbers
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INTEGER VALUED-FUNCTION
INTEGER VALUED-FUNCTION
INTEGER VALUED-FUNCTION
INTEGER VALUED-FUNCTION
INTEGER VALUED-FUNCTION
INTEGER VALUED-FUNCTION
INTEGER VALUED-FUNCTION
INTEGER VALUED-FUNCTION
INTEGER VALUED-FUNCTION
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