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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
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
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
Polynomial with integer value for integer input
mathematics, an integer-valued polynomial (also known as a numerical polynomial) P ( t ) {\displaystyle P(t)} is a polynomial whose value P ( n ) {\displaystyle
Integer-valued_polynomial
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
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 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
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
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 in mathematical number theory
a branch of mathematics, the Carmichael function λ(n) of a positive integer n is the smallest positive integer m such that a m ≡ 1 ( mod n ) {\displaystyle
Carmichael_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
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
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
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
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)
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
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
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
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
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
Function that can be written as a sum over prime factors
an additive function is an arithmetic function f(n) of the positive integer variable n such that whenever a and b are coprime, the function applied to
Additive_function
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
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
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
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)
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)
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
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
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
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
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
number of times a function can be differentiated without producing discontinuities. The smoothness, or differentiability class, is an integer k {\displaystyle
Smoothness
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
Special mathematical function
of positive integer order arise in the calculation of processes represented by higher-order Feynman diagrams. The polylogarithm function is equivalent
Polylogarithm
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
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
Function representing the number of primes less than or equal to a given number
shows how the three functions π(x), x/log x, and li(x) compared at powers of 10. See also, and In the On-Line Encyclopedia of Integer Sequences, the π(x)
Prime-counting_function
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
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
Number-theoretical function
theory, the sum of squares function is an arithmetic function that gives the number of representations for a given positive integer n {\displaystyle n} as
Sum_of_squares_function
Branch of mathematics studying functions of a complex variable
real-valued. In other words, a complex function f : C → C {\displaystyle f:\mathbb {C} \to \mathbb {C} } may be decomposed into two real-valued functions (
Complex_analysis
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
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
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
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
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
Number used for counting
2, 3, and so on, possibly excluding 0. The terms positive integers, non-negative integers, whole numbers, and counting numbers are also used. The set
Natural_number
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
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
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
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
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
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
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
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
Subfield of number theory
represent the ratio of the L-function value to the "transcendental" factor. Subsidiary explanations are given for the integer values of n {\displaystyle n}
Special_values_of_L-functions
Arithmetic operation
integer, the identities are valid for all nonzero complex numbers. If exponentiation is considered as a multivalued function then the possible values
Exponentiation
Solutions of Legendre's differential equation
called the degree and order of the relevant function, respectively. The polynomial solutions when λ is an integer (denoted n), and μ = 0 are the Legendre
Legendre_function
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)
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
Computational operation
a and n both being integers, many computing systems now allow other types of numeric operands. The range of values for an integer modulo operation of
Modulo
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
Complex number whose mapping on a coordinate plane produces a triangular lattice
In mathematics, the Eisenstein integers (named after Gotthold Eisenstein), occasionally also known as Eulerian integers (after Leonhard Euler), are the
Eisenstein_integer
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
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
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
Generalization of the Riemann zeta function for algebraic number fields
Riemann zeta function ζ ( s ) {\displaystyle \zeta (s)} represents information about the factorization of integers. The Dedekind zeta function generalizes
Dedekind_zeta_function
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
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
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
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
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
Computer software bug occurring in 2038
it in a signed 32-bit integer. When the data type's maximum value is exceeded, the integer will overflow to its minimum value, which systems will interpret
Year_2038_problem
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
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
Number systems with a non-integer radix (base), such as base 2.5
non-integer representation uses non-integer numbers as the radix, or base, of a positional numeral system. For a non-integer radix β > 1, the value of
Non-integer base of numeration
Non-integer_base_of_numeration
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
Indicator function of positive numbers
function, or the unit step function, usually denoted by H or θ (but sometimes u, 1 or 𝟙), is a step function named after Oliver Heaviside, the value
Heaviside_step_function
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
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
Fast-growing function
function is a mathematical function defined by Harvey Friedman. It is defined by SSCG ( k ) {\displaystyle {\text{SSCG}}(k)} as the largest integer n
Friedman's_SSCG_function
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
Special mathematical function
(7)\;=\;{\frac {61\pi ^{7}}{184320}}} For the values of the Dirichlet beta function at even positive integers no elementary closed form is known, and no
Dirichlet_beta_function
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
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
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)
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
Finite or infinite ordered list of elements
sequence. A function from Z {\displaystyle \mathbb {Z} } the set of all integers, into a set, for example the sequence of all even integers (..., −4, −2
Sequence
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
Set of rules defining correctly structured programs
interpreted according to use. For example, ⌊3.2 gives 3, the largest integer not above the argument, and 3⌊2 gives 2, the lower of the two arguments
APL_syntax_and_symbols
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)
Algorithm for public-key cryptography
possibility of using Euler totient function results also from Lagrange's theorem applied to the multiplicative group of integers modulo pq. Thus any d satisfying
RSA_cryptosystem
Type of mathematical function
zero. Dirichlet L-functions may be written as linear combinations of the Hurwitz zeta function at rational values. Fixing an integer k ≥ 1 {\displaystyle
Dirichlet_L-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
Function equal to the product of its values on coprime factors
In number theory, a multiplicative function is an arithmetic function f {\displaystyle f} of a positive integer n {\displaystyle n} with the property that
Multiplicative_function
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
Function defined by a hypergeometric series
hypergeometric function adopt a branch cut along the line z ≥ 1. As c → −m, where m is a non-negative integer, one has 2F1(z) → ∞. Dividing by the value Γ(c) of
Hypergeometric_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
INTEGER VALUED-FUNCTION