OEIS - Article Index

Encyclopedia - Alternating Factorial: Encyclopedia - Alternating Factorial
An alternating factorial is the absolute value of the alternating sum of the first n factorials. This is the same as their sum, with the ...   » Read the article

Encyclopedia - Automorphic Number: Encyclopedia - Automorphic Number
In mathematics an automorphic number is a number whose square "ends" in the number itself. For example, 52 = 25, 762 = 5776, and 8906252 ...   » Read the article

Encyclopedia - Bell Number: Encyclopedia - Bell Number
In combinatorial mathematics, the nth Bell number, named in honor of Eric Temple Bell, is the number of partitions of a set with n member...   » Read the article

Encyclopedia - Alkane: Encyclopedia - Alkane
An alkane in organic chemistry is a saturated hydrocarbon without cycles, that is, an acyclic hydrocarbon in which the molecule has the m...   » Read the article

Encyclopedia - Pi: Encyclopedia - Pi
The mathematical constant π is a real number which may be defined as the ratio of a circle's circumference (Greek περιφέρεια, ...   » Read the article

Encyclopedia - Abundant Number: Encyclopedia - Abundant Number
In mathematics, an abundant number or excessive number is a number n for which σ(n) > 2n. Here σ(n) is the divisor function: the sum...   » Read the article

Encyclopedia - N: Encyclopedia - N
N is the fourteenth letter of the modern Latin alphabet. Its name in English is en. Semitic Nûn was probably the picture of a snake; the...   » Read the article

Encyclopedia - Brun's Constant: Encyclopedia - Brun's Constant
In 1919 Viggo Brun showed that the sum of the reciprocals of the twin primes (pairs of prime numbers which differ by 2) converges to a ma...   » Read the article

Encyclopedia - 43 Number: Encyclopedia - 43 Number
43 is the natural number following 42 and preceding 44. << 40 41 42 43 44 45 46 47 48 49 >> List of numbers -- Integers 0 10 ...   » Read the article

Encyclopedia - Catalan Number: Encyclopedia - Catalan Number
In combinatorial mathematics, the Catalan numbers form a sequence of natural numbers that occur in various counting problems; which often...   » Read the article

Encyclopedia - 300 Number: Encyclopedia - 300 Number
Three hundred is the natural number following two hundred and ninety-nine and preceding three hundred one. << 0 100 200 300 400 500...   » Read the article

Encyclopedia - Untouchable Number: Encyclopedia - Untouchable Number
An untouchable number is an integer that can not be expressed as the sum of the proper divisors of any integer. The first few untouchable...   » Read the article

Encyclopedia - Unique Prime: Encyclopedia - Unique Prime
In mathematics, a unique prime is a certain kind of prime number. A prime p ≠ 2, 5 is called unique if there is no other prime q such t...   » Read the article

Encyclopedia - Wilson Prime: Encyclopedia - Wilson Prime
In mathematics, a Wilson prime is a certain kind of prime number. A prime p is called a Wilson prime if p² divides (p − 1)! + 1, where...   » Read the article

Encyclopedia - Woodall Number: Encyclopedia - Woodall Number
In mathematics, a Woodall number is a natural number of the form n · 2n − 1 (written Wn). Woodall numbers were first studied...   » Read the article

Encyclopedia - Constructible Polygon: Encyclopedia - Constructible Polygon
In mathematics, a constructible polygon is a regular polygon that can be constructed with compass and straightedge. For example, a regula...   » Read the article

Encyclopedia - Golden Ratio: Encyclopedia Ii - Golden Ratio - Aesthetic Uses
It has been claimed that the ancient Egyptians knew the golden ratio because ratios close to the golden ratio may be found in the positio...   » Read the article

Encyclopedia - Friedman Number: Encyclopedia Ii - Friedman Number - Algorithms For Finding Friedman Numbers
There usually are fewer 2-digit Friedman numbers than 3-digit and more in any given base, but the 2-digit ones are easier to find. If we ...   » Read the article

Encyclopedia - Lego: Encyclopedia Ii - Lego - Brief History
Main article: History of Lego Also see: Lego timeline The Lego Group had humble beginnings in the workshop of Ole Kirk Christiansen, a po...   » Read the article

Encyclopedia - Magic Square: Encyclopedia Ii - Magic Square - Brief History Of Magic Squares
Magic square - The Lo Shu Square 3x3 magic square. Chinese literature dating from as early as 2800 BC tells the legend of Lo Shu or "sc...   » Read the article

Encyclopedia - Happy Number: Encyclopedia Ii - Happy Number - Sequence Behavior
If n is not happy, then its sequence does not go to 1. What happens instead is that it ends up in the cycle 4, 16, 37, 58, 89, 145, 42, ...   » Read the article

Encyclopedia - E Mathematical Constant: Encyclopedia Ii - E Mathematical Constant - Properties
The exponential function ex is important because it is the unique function (up to multiplication by a constant) which is its own derivati...   » Read the article

Encyclopedia - Harshad Number: Encyclopedia Ii - Harshad Number - What Numbers Can Be Harshad Numbers?
Given the divisibility test for 9, one might be tempted to generalize that all numbers divisible by 9 are also Harshad numbers. But for t...   » Read the article

Encyclopedia - Fibonacci Number: Encyclopedia Ii - Fibonacci Number - Identities
These identities can be proven using many different methods. But, among all, we wish to present an elegant proof for each of them using c...   » Read the article

Encyclopedia - Palindromic Number: Encyclopedia Ii - Palindromic Number - Formal Definition
Although palindromic numbers are most often considered in the decimal system, the concept of palindromicity can be applied to the natural...   » Read the article

Encyclopedia - Cyclic Number: Encyclopedia Ii - Cyclic Number - Special Cases
If leading zeros are not permitted on numerals, then 142857 is the only cyclic number in decimal. Allowing leading zeros, the sequence of...   » Read the article

Encyclopedia - Monstrous Moonshine: Encyclopedia Ii - Monstrous Moonshine - Formal Versions Of Conway's And Norton's Conjectures
The first conjecture made by Conway and Norton was the so-called "moonshine conjecture"; it states that there is an infinite-dimensional ...   » Read the article

Encyclopedia - Magic Constant: Encyclopedia Ii - Magic Constant - Normal Magic Squares
If a magic square of order n is normal (i.e., it contains the numbers 1 to n2), then the magic constant depends only on n; its value is ...   » Read the article

Encyclopedia - Superabundant Number: Encyclopedia Ii - Superabundant Number - Properties
Leonidas Alaoglu and Paul Erdős proved [AlaErd44] that if n is superabundant, then there exist a2, ..., ap such that and In fact, ...   » Read the article

Encyclopedia - Hyperperfect Number: Encyclopedia Ii - Hyperperfect Number - List Of Hyperperfect Numbers
The following table lists the first few k-hyperperfect numbers for some values of k, together with the sequence number in OEIS of the seq...   » Read the article

Encyclopedia - Pi: Encyclopedia Ii - Pi - Numerical Approximations Of π
Due to the transcendental nature of π, there are no closed expressions for the number in terms of algebraic numbers and functions. There...   » Read the article

Encyclopedia - Polyiamond: Encyclopedia Ii - Polyiamond - Counting Polyiamonds
The basic combinatorial question is how many different polyiamonds with a given number of triangles exist. If mirror images are considere...   » Read the article

Encyclopedia - Prime Number: Encyclopedia Ii - Prime Number - Representing Natural Numbers As Products Of Primes
The fundamental theorem of arithmetic states that every positive integer larger than 1 can be written as a product of primes in a unique ...   » Read the article

Encyclopedia - Space Group: Encyclopedia Ii - Space Group - Group Theory
Mathematically, a space group is a symmetry group or symmetry group type of n-dimensional structures with translational symmetry in n ind...   » Read the article

Encyclopedia - Square-free Integer: Encyclopedia Ii - Square-free Integer - Equivalent Characterizations Of Square-free Numbers
The integer n is square-free if and only if in the prime factorization of n, no prime number occurs more than once. Another way of statin...   » Read the article

Encyclopedia - 23 Numerology: Encyclopedia Ii - 23 Numerology - 23 Pattern
There is also a numerical pattern discovered by Anon Castillo using 2, 3, 5, and 23 and including 17, where you add one and two, then add...   » Read the article

Encyclopedia - Square Root Of 2: Encyclopedia Ii - Square Root Of 2 - Proof Of Irrationality
One proof of the number's irrationality is the following proof by infinite descent. It is also a proof by contradiction, which means the ...   » Read the article

Encyclopedia - Perfect Number: Encyclopedia Ii - Perfect Number - Even Perfect Numbers
Euclid discovered that the first four perfect numbers are generated by the formula 2n−1(2n − 1): for n = 2:   21(22 �...   » Read the article

Encyclopedia - Pell Number: Encyclopedia Ii - Pell Number - Pell Numbers
The Pell numbers are defined recursively by: In words: you start with 0 and 1, and then produce the next Pell number by adding twice t...   » Read the article

Encyclopedia - Mersenne Prime: Encyclopedia Ii - Mersenne Prime - Searching For Mersenne Primes
The identity shows that Mn can be prime only if n itself is prime, which simplifies the search for Mersenne primes considerably. The c...   » Read the article

Encyclopedia - Meander Mathematics: Encyclopedia Ii - Meander Mathematics - Meander
Given a fixed oriented line L in the Euclidean plane R2, a meander of order n is a non-self-intersecting closed curve in R2 which transve...   » Read the article

Encyclopedia - Bernoulli Number: Encyclopedia Ii - Bernoulli Number - Introduction
The Bernoulli numbers Bn were first discovered in connection with the closed forms of the sums for various fixed values of n. The clos...   » Read the article

Encyclopedia - Padovan Sequence: Encyclopedia Ii - Padovan Sequence - Recurrence Relations
The Padovan sequence also satisfies the recurrence relations P(n) = P(n − 1) + P(n − 5) P(n) = P(n − 2) + P(n − 4) + P(n − 8) ...   » Read the article

Encyclopedia - 786 Number: Encyclopedia Ii - 786 Number - In Religion
The Arabic letters of the opening phrase of the Qur'an (the Basmala phrase Bismillah al-Rahman al-Rahim "In the name of God, the Compassi...   » Read the article

Encyclopedia - Carmichael Number: Encyclopedia Ii - Carmichael Number - Overview
Fermat's little theorem states that all prime numbers have that property. In this sense, Carmichael numbers are similar to prime numbers....   » Read the article

Encyclopedia - Latin Square: Encyclopedia Ii - Latin Square - Equivalence Classes Of Latin Squares
Many operations on a Latin square produce another Latin square (for example, turning it upside down). If we permute the rows, permute the...   » Read the article

Encyclopedia - Catalan Number: Encyclopedia Ii - Catalan Number - Proof Of The Formula
There are several ways of explaining why the formula given for Cn is correct; that is, why it solves the combinatorial problems listed ab...   » Read the article

Encyclopedia - Bell Number: Encyclopedia Ii - Bell Number - Triangle Scheme For Calculating Bell Numbers
The Bell numbers can easily be calculated by creating the so-called Bell triangle, also called Aitken's array or the Peirce triangle: St...   » Read the article

Encyclopedia - Riemann Zeta Function: Encyclopedia Ii - Riemann Zeta Function - Relationship To Prime Numbers
The connection between this function and prime numbers was already realized by Leonhard Euler: an infinite product extending over all ...   » Read the article

Encyclopedia - Repunit: Encyclopedia Ii - Repunit - Repunit Primes
Historically, the definition of repunits was motivated by recreational mathematicians looking for prime factors of such numbers. Wikipedi...   » Read the article

Encyclopedia - E Mathematical Constant: Encyclopedia Ii - E Mathematical Constant - Properties
The exponential function ex is important because it is the unique function (up to multiplication by a constant) which is its own derivati...   » Read the article

Encyclopedia - Prime Number Theorem: Encyclopedia Ii - Prime Number Theorem - Table Of πx X / Ln X And Lix
Here is a table that shows how the three functions π(x), x / ln x and Li(x) compare: x π(x) π(x) − x / ln x Li(x) − π(x) x / ...   » Read the article

Encyclopedia - Proth's Theorem: Encyclopedia Ii - Proth's Theorem - Numerical Examples
The first seven Proth numbers are (sequence A080075 in OEIS): P0 = 21 + 1 = 3 P1 = 22 + 1 = 5 P2 = 23 + 1 = 9 P3 = 3 × 22 + 1 = 13 P4 =...   » Read the article

Encyclopedia - Alkane: Encyclopedia Ii - Alkane - Molecular Geometry
The molecular structure of the alkanes directly affects their physical and chemical characteristics. It is derived from the electron conf...   » Read the article

Encyclopedia - Sudoku: Encyclopedia Ii - Sudoku - Solution Methods
The strategy for solving a puzzle may be regarded as comprising a combination of three processes: scanning, marking up, and analysing. S...   » Read the article

Encyclopedia - Zeta Constants: Encyclopedia Ii - Zeta Constants - ζ2n+1
By defining the quantities a series of relationships can be given in the form where An,Bn,Cn and Dn are conjectured to be positive ...   » Read the article

Encyclopedia - N: Encyclopedia Ii - N - Usage
N serves as an alveolar nasal in virtually all languages that use the Latin alphabet. A common digraph with N is NG, which produces a vel...   » Read the article

Encyclopedia - Pi: Encyclopedia Ii - Pi - Formulae Involving π
Pi - Geometry. π appears in many formulae in geometry involving circles and spheres. (All of these are a consequence of the first one,...   » Read the article

Encyclopedia - Superior Highly Composite Number: Encyclopedia Ii - Superior Highly Composite Number - Properties
All superior highly composite numbers are highly composite; it can also be shown that there exist prime numbers π1, π2, ... such that t...   » Read the article

Encyclopedia - 43 Number: Encyclopedia Ii - 43 Number - In Mathematics
Forty-three is the 14th smallest prime number. The previous is forty-one, with which it comprises a twin prime, the next is forty-seven. ...   » Read the article

Encyclopedia - 300 Number: Encyclopedia Ii - 300 Number - Other Fields
Three hundred is In bowling, a perfect score, achieved by rolling strikes in all ten frames. The title of a comic book by Frank Miller a...   » Read the article

Encyclopedia - Eight Queens Puzzle: Encyclopedia Ii - Eight Queens Puzzle - The Eight Queens Puzzle As An Exercise In Algorithm Design
Finding all solutions to the eight queens puzzle is a good example of a simple but nontrivial problem. For this reason, it is often used ...   » Read the article

Encyclopedia - Constructible Polygon: Encyclopedia Ii - Constructible Polygon - General Theory
In the light of later work on Galois theory, the principles of these proofs have been clarified. It is straightforward to show from analy...   » Read the article

Encyclopedia - Fermat Number: Encyclopedia Ii - Fermat Number - Primality Of Fermat Numbers
Fermat numbers and Fermat primes were first studied by Pierre de Fermat, who conjectured that all Fermat numbers are prime. Indeed, the f...   » Read the article

Encyclopedia - Factorial: Encyclopedia Ii - Factorial - Factorial-like Products
Factorial - Primorial. The primorial is similar to the factorial, but with the product taken only over the prime numbers. Factorial - ...   » Read the article

Encyclopedia - Constructible Polygon: Encyclopedia Ii - Constructible Polygon - Conditions For Constructibility
Some regular polygons are easy to construct with compass and straightedge; others are not. This led to the question being posed: is it po...   » Read the article

Encyclopedia - Square-free Integer: Encyclopedia Ii - Square-free Integer - Distribution Of Square-free Numbers
If Q(x) denotes the number of square-free integers between 1 and x, then (see pi and big O notation). The asymptotic/natural density o...   » Read the article

Encyclopedia - Riemann Zeta Function: Encyclopedia Ii - Riemann Zeta Function - Applications
Although mathematicians regard the Riemann zeta function as being primarily relevant to the "purest" of mathematical disciplines, number ...   » Read the article

Encyclopedia - Square Root Of 2: Encyclopedia Ii - Square Root Of 2 - History
The first approximation of this number was given in ancient Indian mathematical texts, the Sulbasutras (800 B.C. to 200 B.C.) as follows:...   » Read the article

Encyclopedia - Latin Square: Encyclopedia Ii - Latin Square - Latin Squares And Mathematical Puzzles
The popular Sudoku puzzles are a special case of Latin squares; any solution to a Sudoku puzzle is a Latin square. Sudoku imposes the add...   » Read the article

Encyclopedia - Space Group: Encyclopedia Ii - Space Group - Space Groups In Crystallography
The space groups in three dimensions are made from combinations of the 32 crystallographic point groups with the 14 Bravais lattices whic...   » Read the article

Encyclopedia - Riemann Zeta Function: Encyclopedia Ii - Riemann Zeta Function - Series Expansions
The Riemann zeta function is meromorphic with a single pole of order one at s = 1. It can therefore be expanded as a Laurent se...   » Read the article

Encyclopedia - Riemann Zeta Function: Encyclopedia Ii - Riemann Zeta Function - Basic Properties
The zeta function satisfies the following functional equation: valid for all s in C\{0,1}. Here, Γ denotes the gamma function. This f...   » Read the article

Encyclopedia - Pi: Encyclopedia Ii - Pi - History Of π
Main article: History of Pi. The value of π has been known in some form since antiquity. As early as the 20th century BC, Babylonian mat...   » Read the article

Encyclopedia - Pi: Encyclopedia Ii - Pi - Formulae Involving π
Pi - Geometry. π appears in many formulae in geometry involving circles and spheres. (All of these are a consequence of the first one,...   » Read the article

Encyclopedia - Riemann Zeta Function: Encyclopedia Ii - Riemann Zeta Function - The Riemann Zeta Function As A Mellin Transform
The Mellin transform of a function f(x) is defined as in the region where the integral is defined. There are various expressions for t...   » Read the article

Encyclopedia - Pi: Encyclopedia Ii - Pi - Properties
π is an irrational number; that is, it cannot be written as the ratio of two integers, as was proven in 1761 by Johann Heinrich Lambert....   » Read the article

Encyclopedia - Riemann Zeta Function: Encyclopedia Ii - Riemann Zeta Function - Generalizations
There are a number of related zeta functions that can be considered to be generalizations of Riemann's zeta. The simplest of these are th...   » Read the article

Encyclopedia - Prime Number Theorem: Encyclopedia Ii - Prime Number Theorem - The Prime Counting Function In Terms Of The Logarithmic Integral
Gauss conjectured that an even better approximation to π(x) is given by the offset logarithmic integral function Li(x), defined by In...   » Read the article

Encyclopedia - Monstrous Moonshine: Encyclopedia Ii - Monstrous Moonshine - Borcherds' Proof
Richard Ewen Borcherds' proof of the conjecture of Conway and Norton can be broken into five major steps as follows: A vertex algebra V ...   » Read the article

Encyclopedia - Sudoku: Encyclopedia Ii - Sudoku - History
The puzzle was designed by Howard Garns, a retired architect and freelance puzzle constructor, and first published in 1979. Although like...   » Read the article

Encyclopedia - Zeta Constants: Encyclopedia Ii - Zeta Constants - ζ5
Simon Plouffe gives the identities and ... See also:Zeta constants, Zeta constants - ζ3, Zeta constants - ζ5, Zeta constants - ...   » Read the article

Encyclopedia - Palindromic Number: Encyclopedia Ii - Palindromic Number - Decimal Palindromic Numbers
All numbers in base 10 with one digit {0, 1, 2, 3, 4, 5, 6, 7, 8, 9} are palindromic ones. The number of palindromic numbers with two dig...   » Read the article

Encyclopedia - Palindromic Number: Encyclopedia Ii - Palindromic Number - Other Bases
Palindromic numbers can be considered in other numeral systems than decimal. For example, the binary palindromic numbers are: 0, 1, 11, ...   » Read the article

Encyclopedia - Sudoku: Encyclopedia Ii - Sudoku - Mathematics Of Sudoku
The general problem of solving Sudoku puzzles on n2 x n2 boards of n x n blocks is known to be NP-complete [6]. This gives some indicatio...   » Read the article

Encyclopedia - Sudoku: Encyclopedia Ii - Sudoku - Variants
Although the 9×9 grid with 3×3 regions is by far the most common, numerous variations abound: sample puzzles can be 4×4 grids with 2×...   » Read the article

Encyclopedia - Latin Square: Encyclopedia Ii - Latin Square - Examples
We give one example of a Latin square from each main class up to order 5. They present, respectively, the multiplication tables of the ...   » Read the article

Encyclopedia - Sudoku: Encyclopedia Ii - Sudoku - Difficulty Ratings
Published puzzles often are ranked in terms of difficulty. Perhaps surprisingly, the number of givens has little or no bearing on a puzzl...   » Read the article

Encyclopedia - Prime Number Theorem: Encyclopedia Ii - Prime Number Theorem - Statement Of The Theorem
Let π(x) be the prime counting function that gives the number of primes less than or equal to x, for any real number x. For example, π(...   » Read the article

Encyclopedia - Sudoku: Encyclopedia Ii - Sudoku - Construction
It is possible to set starting grids with more than one solution and to set grids with no solution, but such are not considered proper Su...   » Read the article

Encyclopedia - Sudoku: Encyclopedia Ii - Sudoku - Introduction
The name Sudoku is the Japanese abbreviation of a longer phrase, "suji wa dokushin ni kagiru," meaning "the digits must remain single"; i...   » Read the article

Encyclopedia - Riemann Zeta Function: Encyclopedia Ii - Riemann Zeta Function - Definition
The Riemann zeta function ζ(s) is defined for any complex number s with real part > 1 by the Dirichlet series: In the region {s in...   » Read the article

Encyclopedia - Cyclic Number: Encyclopedia Ii - Cyclic Number - Relation To Recurring Decimals
The Cyclic numbers are related to the repeating digital representations of unit fractions. In general, for a cyclic number of length L, t...   » Read the article

Encyclopedia - Cyclic Number: Encyclopedia Ii - Cyclic Number - Form Of Cyclic Numbers
From the relation to unit fractions, it can be shown that cyclic numbers are of the form where b is the number base (10 for decimal), ...   » Read the article

Encyclopedia - Prime Number Theorem: Encyclopedia Ii - Prime Number Theorem - Bounds On The Prime Counting Function
The prime number theorem is an asymptotic result. Hence, it cannot be used to bound π(x). However, some bounds on π(x) are known, for i...   » Read the article

Encyclopedia - Prime Number Theorem: Encyclopedia Ii - Prime Number Theorem - The Prime Number Theorem For Arithmetic Progressions
Let πn,a(x) denote the number of primes in the arithmetic progression a, a + n, a + 2n, a + 3n, … less than x. Dirichlet and Legendre ...   » Read the article

Encyclopedia - Carmichael Number: Encyclopedia Ii - Carmichael Number - Higher-order Carmichael Numbers
Carmichael numbers can be generalized using concepts of abstract algebra. The above definition states that a composite integer n is Carmi...   » Read the article

Encyclopedia - Cyclic Number: Encyclopedia Ii - Cyclic Number - Construction Of Cyclic Numbers
Cyclic numbers can be constructed by the following procedure: Let b be the number base (10 for decimal) Let p be a prime that does not di...   » Read the article

Encyclopedia - Cyclic Number: Encyclopedia Ii - Cyclic Number - Other Numeric Bases
Using the above technique, cyclic numbers can be found in other numeric bases. In binary, the sequence of cyclic numbers begins: 01 0011...   » Read the article

Encyclopedia - Fibonacci Number: Encyclopedia Ii - Fibonacci Number - Applications
The Fibonacci numbers are important in the run-time analysis of Euclid's algorithm to determine the greatest common divisor of two intege...   » Read the article

Encyclopedia - Prime Number Theorem: Encyclopedia Ii - Prime Number Theorem - Gaps Between Primes
The prime number theorem says that the "average" length of the gap between a prime p and the next prime is ln p. Of course, the actu...   » Read the article