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Function whose domain is the positive integers
\log _{e}(x)} . In number theory, an arithmetic, arithmetical, or number-theoretic function is generally any function whose domain is the set of positive
Arithmetic_function
System of arithmetic in proof theory
elementary function arithmetic (EFA), also called elementary arithmetic and exponential function arithmetic, is the system of arithmetic with the usual
Elementary function arithmetic
Elementary_function_arithmetic
Function defined on integers in number theory
In number theory, the Lagarias arithmetic derivative or number derivative is a function defined for integers, based on prime factorization, by analogy
Arithmetic_derivative
Multiplicative function in number theory
the OEIS). In number theory another arithmetic function closely related to the Möbius function is the Mertens function, defined by M ( n ) = ∑ k = 1 n μ
Möbius_function
Type of zeta function
mathematics, the arithmetic zeta function is a zeta function associated with a scheme of finite type over integers. The arithmetic zeta function generalizes
Arithmetic_zeta_function
In number theory, the gcd-sum function, also called Pillai's arithmetical function, is defined for every n {\displaystyle n} by P ( n ) = ∑ k = 1 n gcd
Pillai's arithmetical function
Pillai's_arithmetical_function
Arithmetic function
multiplicative function (or totally multiplicative function) is an arithmetic function (that is, a function whose domain is the positive integers), such that
Completely multiplicative function
Completely_multiplicative_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
arithmetic function is some simpler or better-understood function which takes the same values "on average". Let f {\displaystyle f} be an arithmetic function
Average order of an arithmetic function
Average_order_of_an_arithmetic_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
Method for bounding the errors of numerical computations
errors in mathematical computation by computing function bounds. Numerical methods involving interval arithmetic can guarantee relatively reliable and mathematically
Interval_arithmetic
Function on an integer n which is log(p) if n equals p^k and zero otherwise
Mangoldt function is an arithmetic function named after German mathematician Hans von Mangoldt. It is an example of an important arithmetic function that
Von_Mangoldt_function
Type of asymptotic behavior useful in number theory
arithmetic function is some simpler or better-understood function which "usually" takes the same or closely approximate values. Let f be a function on
Normal order of an arithmetic function
Normal_order_of_an_arithmetic_function
Number-theoretical function
In number theory, the sum of squares function is an arithmetic function that gives the number of representations for a given positive integer n {\displaystyle
Sum_of_squares_function
orders of an arithmetic function in number theory, a branch of mathematics, are the best possible bounds of the given arithmetic function. Specifically
Extremal orders of an arithmetic function
Extremal_orders_of_an_arithmetic_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
Number of integers coprime to and less than n
Dirichlet's theorem on arithmetic progressions. The Dirichlet series for φ(n) may be written in terms of the Riemann zeta function as: ∑ n = 1 ∞ φ ( n )
Euler's_totient_function
Mathematical operation on arithmetical functions
convolution (or divisor convolution) is a binary operation defined for arithmetic functions; it is important in number theory. It was developed by Peter Gustav
Dirichlet_convolution
Formula for the sum of an arithmetic function
sum of an arithmetic function, by means of an inverse Mellin transform. Let { a ( n ) } {\displaystyle \{a(n)\}} be an arithmetic function, and let g
Perron's_formula
Branch of algebraic geometry
Rational points can be directly characterized by height functions which measure their arithmetic complexity. The structure of algebraic varieties defined
Arithmetic_geometry
Number taken as representative of a list of numbers
its overall position. In mathematics, it most commonly refers to the arithmetic mean, but may also refer to other measures such as other types of mean
Average
Function representing the number of primes less than or equal to a given number
(t)}{t\log ^{2}(t)}}\mathrm {d} t.} Formulas for prime-counting functions come in two kinds: arithmetic formulas and analytic formulas. Analytic formulas for prime-counting
Prime-counting_function
Number of partitions of an integer
is credited with discovering that the partition function has nontrivial patterns in modular arithmetic. For instance the number of partitions is divisible
Partition function (number theory)
Partition_function_(number_theory)
Number divisible only by 1 and itself
there are arbitrarily long finite arithmetic progressions consisting only of primes. Euler noted that the function n 2 − n + 41 {\displaystyle n^{2}-n+41}
Prime_number
Topics referred to by the same term
by sigma function one can mean one of the following: The sum-of-divisors function σa(n), an arithmetic function Weierstrass sigma function, related to
Sigma_function
arithmetic function over the divisors of a natural number n {\displaystyle n} , or equivalently the Dirichlet convolution of an arithmetic function f
Divisor_sum_identities
Branch of pure mathematics
of mathematics devoted primarily to the study of the integers and arithmetic functions. Number theorists study prime numbers as well as the properties of
Number_theory
Arithmetic function
Liouville function, named after French mathematician Joseph Liouville and denoted λ ( n ) {\displaystyle \lambda (n)} , is an important arithmetic function. Its
Liouville_function
Romanian mathematician
equations, linear recurrences and the distribution of values of arithmetic functions. He has made notable contributions to the proof that irrational automatic
Florian_Luca
Type of average of a collection of numbers
In mathematics and statistics, the arithmetic mean ( /ˌærɪθˈmɛtɪk/ arr-ith-MET-ik), arithmetic average, or just the mean or average is the sum of a collection
Arithmetic_mean
Computation modulo a fixed integer
In mathematics, modular arithmetic is a system of arithmetic operations for integers, differing from the usual ones in that numbers "wrap around" when
Modular_arithmetic
Integers have unique prime factorizations
In mathematics, the fundamental theorem of arithmetic, also called the unique factorization theorem and prime factorization theorem, states that every
Fundamental theorem of arithmetic
Fundamental_theorem_of_arithmetic
Relation between pairs of arithmetic functions
classic Möbius inversion formula is a relation between pairs of arithmetic functions, each defined from the other by sums over divisors. It was introduced
Möbius_inversion_formula
Topics referred to by the same term
Fourier coefficients of the Ramanujan modular form Divisor function, an arithmetic function giving the number of divisors of an integer This disambiguation
Tau_function
Arithmetic function
\mathbb {P} }\left(1+{\frac {1}{p^{2}(p-1)}}\right)=1.339784\ldots .} Arithmetic function Graham, Ronald L.; Knuth, Donald E.; Patashnik, Oren. Concrete Mathematics
Totient_summatory_function
IEEE standard for floating-point arithmetic
numbers during arithmetic and conversions operations: arithmetic and other operations (such as trigonometric functions) on arithmetic formats exception
IEEE_754
Iterative algorithm on numbers
_{i=0}^{n}b^{i}\right)+k\\&=m\\\end{aligned}}} Mathematics portal Arithmetic dynamics Collatz conjecture Dudeney number Factorion Happy number Kaprekar
Kaprekar's_routine
Mathematical function
the Chebyshev function is either a scalarising function (Tchebycheff function) or one of two related functions. The first Chebyshev function ϑ(x) or θ(x)
Chebyshev_function
function Mathieu function Mittag-Leffler function Painlevé transcendents Parabolic cylinder function Arithmetic–geometric mean Ackermann function: in the theory
List of mathematical functions
List_of_mathematical_functions
Sum of all proper divisors of a natural number
26, 1, 76, 8, 43, ... (sequence A001065 in the OEIS) The aliquot sum function can be used to characterize several notable classes of numbers: 1 is the
Aliquot_sum
Type of computer arithmetic
algorithms for arithmetic operations, were introduced by Charles Clenshaw and Frank Olver in 1984. The symmetric form of the LI system and its arithmetic operations
Symmetric level-index arithmetic
Symmetric_level-index_arithmetic
Figurate number
S2CID 53079729 Wikimedia Commons has media related to triangular numbers. "Arithmetic series", Encyclopedia of Mathematics, EMS Press, 2001 [1994] Triangular
Triangular_number
general towers. A p-adic L-function arising in this way is typically called an arithmetic p-adic L-function as it encodes arithmetic data of the Galois module
P-adic_L-function
Integer having a non-trivial divisor
order of the factors. This fact is called the fundamental theorem of arithmetic. There are several known primality tests that can determine whether a
Composite_number
Product of two prime numbers
Sequences. OEIS Foundation. Nowicki, Andrzej (2013-07-01), Second numbers in arithmetic progressions, arXiv:1306.6424 Conway, J. H. (2008-06-18), Counting Groups:
Semiprime
Number used for counting
numbers coming after smaller ones in the list 1, 2, 3, .... Two basic arithmetical operations are defined on natural numbers: addition and multiplication
Natural_number
Axioms for the natural numbers
define the arithmetical properties of the natural numbers. The naturals are assumed to be closed under a single-valued "successor" function S. For every
Peano_axioms
Mathematical operation
series, or Dirichlet generating function (DGF), of a sequence is a common way of understanding and summing arithmetic functions in a meaningful way. A little
Dirichlet_series_inversion
Numbers obtained by adding the two previous ones
} For a given n, this matrix can be computed in O(log n) arithmetic operations, using the exponentiation by squaring method. Taking the determinant
Fibonacci_sequence
Replacing a number with a simpler value
when dividing two numbers in integer or fixed-point arithmetic; when computing mathematical functions such as square roots, logarithms, and sines; or when
Rounding
Mathematical series
(n)}{n^{s}}}} where L(χ, s) is a Dirichlet L-function. If the arithmetic function f has a Dirichlet inverse function f − 1 ( n ) {\displaystyle f^{-1}(n)}
Dirichlet_series
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 function
Integer_function
Recursive integer sequence
binomial coefficients, by Stirling's approximation for n!, or via generating functions. The only Catalan numbers Cn that are odd are those for which n = 2k −
Catalan_number
Combinational digital circuit
In computing, an arithmetic logic unit (ALU) is a combinational digital circuit that performs arithmetic and bitwise operations on integer binary numbers
Arithmetic_logic_unit
Number of prime factors of a natural number
counts the total number of prime factors with multiplicity (see arithmetic function). That is, if we have a prime factorization of n {\displaystyle n}
Prime_omega_function
Natural number
1088/0026-1394/31/6/013. Peano, Giuseppe (1889). Arithmetices principia, nova methodo exposita [The principles of arithmetic, presented by a new method]. An excerpt
1
Summatory function of the Möbius function
Perron's formula Liouville's function Davenport, H. (November 1937). "On Some Infinite Series Involving Arithmetical Functions (Ii)". The Quarterly Journal
Mertens_function
Mathematical system
In mathematical logic, second-order arithmetic is a collection of axiomatic systems that formalize the natural numbers and their subsets. It is an alternative
Second-order_arithmetic
Branch of mathematical logic
common in second-order arithmetic, is greatly reduced. For example, a continuous function on the Cantor space is just a function that maps binary sequences
Reverse_mathematics
Class of natural numbers with many divisors
{d(n)}{n^{\varepsilon }}}\geq {\frac {d(k)}{k^{\varepsilon }}}} where d(n), the divisor function, denotes the number of divisors of n. The term was coined by Ramanujan
Superior highly composite number
Superior_highly_composite_number
Well-quasi-ordering of finite trees
arithmetic grows phenomenally fast as a function of n {\displaystyle n} , far faster than any primitive recursive function or the Ackermann function,
Kruskal's_tree_theorem
Arithmetic operation
operation with integer exponents may be defined directly from elementary arithmetic operations. The definition of the exponentiation as an iterated multiplication
Exponentiation
Conjecture on zeros of the zeta function
every arithmetic scheme or a scheme of finite type over integers. The arithmetic zeta function of a regular connected equidimensional arithmetic scheme
Riemann_hypothesis
Hardware description language (VHDL) library package for use in electronic circuit design
for VHDL. It provides arithmetic functions for vectors. Overrides of std_logic_vector are defined for signed and unsigned arithmetic. It defines numeric
Numeric_std
Triangular array of natural numbers
N ( n , k ) {\displaystyle \operatorname {N} (n,k)} . The generating function for the Narayana numbers is ∑ n = 1 ∞ ∑ k = 1 n N ( n , k ) z n t k −
Narayana_number
Mathematical function
In mathematics, Landau's function g(n), named after Edmund Landau, is defined for every natural number n to be the largest order of an element of the symmetric
Landau's_function
Generalization of means
quasi-arithmetic mean or generalised f-mean or Kolmogorov-Nagumo-de Finetti mean is one generalisation of the more familiar means such as the arithmetic mean
Quasi-arithmetic_mean
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
Summability method in physics
{\displaystyle \epsilon _{i,j,k}} Zeta-function regularization gives an analytic structure to any sums over an arithmetic function f(n). Such sums are known as
Zeta_function_regularization
Count of the possible partitions of a set
doi:10.1017/S1757748900002334. Becker, H. W.; Riordan, John (1948). "The arithmetic of Bell and Stirling numbers". American Journal of Mathematics. 70 (2):
Bell_number
Function in number theory given by Srinivasa Ramanujan
Spilker, Jürgen (1994). Arithmetical Functions. An introduction to elementary and analytic properties of arithmetic functions and to some of their almost-periodic
Ramanujan's_sum
Computer approximation for real numbers
In computing, floating-point arithmetic (FP) is arithmetic on subsets of real numbers formed by a significand (a signed sequence of a fixed number of
Floating-point_arithmetic
Summatory function of the divisor-counting 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
Divisor_summatory_function
Binary representation for signed numbers
Affeldt, Reynald & Marti, Nicolas (2006). Formal verification of arithmetic functions in SmartMIPS Assembly (PDF) (Report). Archived from the original
Two's_complement
set one of its elements. These properties concern how the function is affected by arithmetic operations on its argument. The following are special examples
List_of_types_of_functions
Numbers whose prime factors all divide the number more than once
(2k+1 − 1)k+1 are k-powerful numbers in an arithmetic progression. Moreover, if a1, a2, ..., as are k-powerful in an arithmetic progression with common difference
Powerful_number
Device used for calculations
portable electronic device used to perform calculations, ranging from basic arithmetic to complex mathematics. The first solid-state electronic calculator was
Calculator
Product of an integer with itself
is the difference-of-squares formula, which can be useful for mental arithmetic: for example, 47 × 53 can be easily computed as 502 − 32 = 2500 − 9 =
Square_number
Number that when multiplied by another number moves its last digit to its front
pseudoprime Strong pseudoprime Arithmetic functions and dynamics Divisor functions Abundant Almost perfect Arithmetic Betrothed Colossally abundant Deficient
Parasitic_number
following outline is provided as an overview of and topical guide to arithmetic: Arithmetic is an elementary branch of mathematics that deals with numerical
Outline_of_arithmetic
Number that is the result of operation on its own digits
expression using all its own digits in combination with any of the four basic arithmetic operators (+, −, ×, ÷), additive inverses, parentheses, exponentiation
Friedman_number
Number
consequently dividing by 0 is generally considered to be undefined in arithmetic. As a numerical digit, 0 plays a crucial role in decimal notation: it
0
Formalization of the natural numbers
see Skolem arithmetic. The language of PRA can express arithmetic propositions involving natural numbers and any primitive recursive function, including
Primitive recursive arithmetic
Primitive_recursive_arithmetic
Complex-valued arithmetic function
number theory and related branches of mathematics, a complex-valued arithmetic function χ : Z → C {\displaystyle \chi :\mathbb {Z} \rightarrow \mathbb {C}
Dirichlet_character
Mathematical concept
the arithmetic mean of the limit from the left and the limit from the right at discontinuities. His formula was given in terms of the related function f
Explicit formulae for L-functions
Explicit_formulae_for_L-functions
study properties of arithmetical functions. Bell series were introduced and developed by Eric Temple Bell. Given an arithmetic function f {\displaystyle
Bell_series
Certain type of divisor of an integer
[The number of bi-unitary divisors of an integer, in The Theory of Arithmetic Functions, Lecture Notes in Mathematics 251: 273–282, New York, Springer–Verlag]
Unitary_divisor
Dirichlet's theorem on arithmetic progressions Linnik's theorem Elliott–Halberstam conjecture Functional equation (L-function) Chebotarev's density theorem
List_of_number_theory_topics
Quickly growing function
total-computable-but-not-primitive-recursive function, Ackermann's original function is seen to extend the basic arithmetic operations beyond exponentiation, although
Ackermann_function
Ten raised to an integer power
pseudoprime Strong pseudoprime Arithmetic functions and dynamics Divisor functions Abundant Almost perfect Arithmetic Betrothed Colossally abundant Deficient
Power_of_10
Limitative results in mathematical logic
provability within a system could be expressed purely in terms of arithmetical functions that operate on Gödel numbers of sentences of the system. Therefore
Gödel's incompleteness theorems
Gödel's_incompleteness_theorems
Numbers with a certain property involving recursive summation
episode "42", a sequence of happy primes is the password to open a door. Arithmetic dynamics Fortunate number Harshad number Lucky number Perfect digital
Happy_number
Integer where the average of its positive divisors is also an integer
theory, an arithmetic number is an integer for which the average of its positive divisors is also an integer. For instance, 6 is an arithmetic number because
Arithmetic_number
Theories in mathematical logic
fragments of Peano arithmetic. The case n = 1 has about the same strength as primitive recursive arithmetic (PRA). Exponential function arithmetic (EFA) is IΣ0
List_of_first-order_theories
S-shaped curve
logarithmic curve, and by analogy with arithmetic and geometric. His growth model is preceded by a discussion of arithmetic growth and geometric growth (whose
Logistic_function
Hierarchy of complexity classes for formulas defining sets
In mathematical logic, the arithmetical hierarchy, arithmetic hierarchy or Kleene–Mostowski hierarchy (after mathematicians Stephen Cole Kleene and Andrzej
Arithmetical_hierarchy
Integer filtered out using a sieve similar to that of Eratosthenes
pseudoprime Strong pseudoprime Arithmetic functions and dynamics Divisor functions Abundant Almost perfect Arithmetic Betrothed Colossally abundant Deficient
Lucky_number
Mathematical function of two positive real arguments
geometric means. The arithmetic–geometric mean is used in fast algorithms for exponential, trigonometric functions, and other special functions, as well as some
Arithmetic–geometric_mean
Topics referred to by the same term
operator that commutes with its Hermitian adjoint Normal order of an arithmetic function, a type of asymptotic behavior useful in number theory Normal polytopes
Normal
Number raised to the third power
In arithmetic and algebra, the cube of a number n is its third power, that is, the result of multiplying three instances of n together. The cube of a number
Cube_(algebra)
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