|
115 (number)
115 (one hundred ndfifteen) is the natural number following 114 and preceding 116. In mathematics 115 has a square sum of divisors: :\sigma(115)=1+5+23+115=144=12^2. There are 115 different rooted trees with exactly eight nodes, 115 inequivalent ways of placing six rooks on a 6 × 6 chess board in such a way that no two of the rooks attack each other, and 115 solutions to the stamp folding problem for a strip of seven stamps. 115 is also a heptagonal pyramidal number. The 115th Woodall number, :115\cdot 2^-1=4\;776\;913\;109\;852\;041\;418\;248\;056\;622\;882\;488\;319, is a prime number. 115 is the sum of the first five heptagonal numbers A heptagonal number is a figurate number that is constructed by combining heptagons with ascending size. The ''n''-th heptagonal number is given by the formula :H_n=\frac. The first few heptagonal numbers are: : 0, 1, 7, 18, 34, 55, 81, 112 .... See also * 115 (other) References Integers {{Use dm ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Natural Number
In mathematics, the natural numbers are those numbers used for counting (as in "there are ''six'' coins on the table") and ordering (as in "this is the ''third'' largest city in the country"). Numbers used for counting are called ''Cardinal number, cardinal numbers'', and numbers used for ordering are called ''Ordinal number, ordinal numbers''. Natural numbers are sometimes used as labels, known as ''nominal numbers'', having none of the properties of numbers in a mathematical sense (e.g. sports Number (sports), jersey numbers). Some definitions, including the standard ISO/IEC 80000, ISO 80000-2, begin the natural numbers with , corresponding to the non-negative integers , whereas others start with , corresponding to the positive integers Texts that exclude zero from the natural numbers sometimes refer to the natural numbers together with zero as the whole numbers, while in other writings, that term is used instead for the integers (including negative integers). The natural ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
114 (number)
114 (one hundred ndfourteen) is the natural number following 113 and preceding 115. In mathematics *114 is an abundant number, a sphenic number and a Harshad number. It is the sum of the first four hyperfactorials, including H(0). At 114, the Mertens function sets a new low of -6, a record that stands until 197. *114 is the smallest positive integer* which has yet to be represented as a3 + b3 + c3, where a, b, and c are integers. It is conjectured that 114 can be represented this way. (*Excluding integers of the form 9k ± 4, for which solutions are known not to exist.) *There is no answer to the equation φ(x) = 114, making 114 a nontotient. *114 appears in the Padovan sequence, preceded by the terms 49, 65, 86 (it is the sum of the first two of these). *114 is a repdigit In recreational mathematics, a repdigit or sometimes monodigit is a natural number composed of repeated instances of the same digit in a positional number system (often implicitly decimal). The word is a po ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
116 (number)
116 (one hundred ndsixteen) is the natural number following 115 and preceding 117. In mathematics 116 is a noncototient, meaning that there is no solution to the equation , where stands for Euler's totient function. 116! + 1 is a factorial prime. There are 116 ternary Lyndon words of length six, and 116 irreducible polynomials of degree six over a three-element field, which form the basis of a free Lie algebra of dimension 116. There are 116 different ways of partitioning the numbers from 1 through 5 into subsets in such a way that, for every ''k'', the union of the first ''k'' subsets is a consecutive sequence of integers. There are 116 different 6×6 Costas arrays.. See also *116 (other) 116 (''one hundred and sixteen'') may refer to: *116 (number) *AD 116 *116 BC *116 (Devon and Cornwall) Engineer Regiment, Royal Engineers, a military unit *116 (MBTA bus) *116 (New Jersey bus) *116 (hip hop group), a Christian hip hop collective * ... References {{DEFAULT ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Square Number
In mathematics, a square number or perfect square is an integer that is the square (algebra), square of an integer; in other words, it is the multiplication, product of some integer with itself. For example, 9 is a square number, since it equals and can be written as . The usual notation for the square of a number is not the product , but the equivalent exponentiation , usually pronounced as " squared". The name ''square'' number comes from the name of the shape. The unit of area is defined as the area of a unit square (). Hence, a square with side length has area . If a square number is represented by ''n'' points, the points can be arranged in rows as a square each side of which has the same number of points as the square root of ''n''; thus, square numbers are a type of figurate numbers (other examples being Cube (algebra), cube numbers and triangular numbers). Square numbers are non-negative. A non-negative integer is a square number when its square root is again an intege ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rooted Tree
In graph theory, a tree is an undirected graph in which any two vertices are connected by ''exactly one'' path, or equivalently a connected acyclic undirected graph. A forest is an undirected graph in which any two vertices are connected by ''at most one'' path, or equivalently an acyclic undirected graph, or equivalently a disjoint union of trees. A polytreeSee . (or directed tree or oriented treeSee .See . or singly connected networkSee .) is a directed acyclic graph (DAG) whose underlying undirected graph is a tree. A polyforest (or directed forest or oriented forest) is a directed acyclic graph whose underlying undirected graph is a forest. The various kinds of data structures referred to as trees in computer science have underlying graphs that are trees in graph theory, although such data structures are generally rooted trees. A rooted tree may be directed, called a directed rooted tree, either making all its edges point away from the root—in which case it is called an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rook (chess)
The rook (; ♖, ♜) is a piece in the game of chess. It may move any number of squares horizontally or vertically without jumping, and it may an enemy piece on its path; additionally, it may participate in castling. Each player starts the game with two rooks, one in each corner on their own side of the board. Formerly, the rook (from Persian رخ ''rokh''/''rukh'', meaning "chariot") was alternatively called the tower, marquess, rector, and comes (count or earl). The term "castle" is considered to be informal, incorrect, or old-fashioned. Placement and movement The white rooks start on squares a1 and h1, while the black rooks start on a8 and h8. The rook moves horizontally or vertically, through any number of unoccupied squares (see diagram). The rook cannot jump over pieces. The rook may capture an enemy piece by moving to the square on which the enemy piece stands, removing it from play. The rook also participates with the king in a special move called castling, wherein i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Chess Board
A chessboard is a used to play chess. It consists of 64 squares, 8 rows by 8 columns, on which the chess pieces are placed. It is square in shape and uses two colours of squares, one light and one dark, in a chequered pattern. During play, the board is oriented such that each player's near-right corner square is a light square. The columns of a chessboard are known as ', the rows are known as ', and the lines of adjoining same-coloured squares (each running from one edge of the board to an adjacent edge) are known as '. Each square of the board is named using algebraic, descriptive, or numeric chess notation; algebraic notation is the FIDE standard. In algebraic notation, using White's perspective, files are labeled ''a'' through ''h'' from left to right, and ranks are labeled ''1'' through ''8'' from bottom to top; each square is identified by the file and rank which it occupies. The a- through d-files comprise the , while the e- through h-files comprise the . History and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Map Folding
In the mathematics of paper folding, map folding and stamp folding are two problems of counting the number of ways that a piece of paper can be folded. In the stamp folding problem, the paper is a strip of stamps with creases between them, and the folds must lie on the creases. In the map folding problem, the paper is a map, divided by creases into rectangles, and the folds must again lie only along these creases. credits the invention of the stamp folding problem to Émile Lemoine. provides several other early references. Labeled stamps In the stamp folding problem, the paper to be folded is a strip of square or rectangular stamps, separated by creases, and the stamps can only be folded along those creases. In one commonly considered version of the problem, each stamp is considered to be distinguishable from each other stamp, so two foldings of a strip of stamps are considered equivalent only when they have the same vertical sequence of stamps. For example, there are six ways to ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Heptagonal Pyramidal Number
A pyramidal number is a figurate number that represents a pyramid with a polygonal base and a given number of triangular sides. A pyramidal number is the number of points in a pyramid where each layer of the pyramid is an -sided polygon of points. The term often refers to square pyramidal numbers, which have a square base with four sides, but it can also refer to pyramids with three or more sides. The numbers of points in the base (and in parallel layers to the base) are given by polygonal numbers of the given number of sides, while the numbers of points in each triangular side is given by a triangular number. It is possible to extend the pyramidal numbers to higher dimensions. Formula The formula for the th -gonal pyramidal number is :P_n^r= \frac, where , . This formula can be factored: :P_n^r=\frac=\left(\frac\right)\left(\frac\right)=T_n \cdot \frac, where is the th triangular number. Sequences The first few triangular pyramidal numbers (equivalently, tetrahedral numb ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Woodall Number
In number theory, a Woodall number (''W''''n'') is any natural number of the form :W_n = n \cdot 2^n - 1 for some natural number ''n''. The first few Woodall numbers are: :1, 7, 23, 63, 159, 383, 895, … . History Woodall numbers were first studied by Allan J. C. Cunningham and H. J. Woodall in 1917, inspired by James Cullen's earlier study of the similarly defined Cullen numbers. Woodall primes Woodall numbers that are also prime numbers are called Woodall primes; the first few exponents ''n'' for which the corresponding Woodall numbers ''W''''n'' are prime are 2, 3, 6, 30, 75, 81, 115, 123, 249, 362, 384, ... ; the Woodall primes themselves begin with 7, 23, 383, 32212254719, ... . In 1976 Christopher Hooley showed that almost all Cullen numbers are composite. In October 1995, Wilfred Keller published a paper discussing several new Cullen primes and the efforts made to factorise other Cullen and Woodall numbers. Included in that paper is a personal communication to Kel ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Prime Number
A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways of writing it as a product, or , involve 5 itself. However, 4 is composite because it is a product (2 × 2) in which both numbers are smaller than 4. Primes are central in number theory because of the fundamental theorem of arithmetic: every natural number greater than 1 is either a prime itself or can be factorized as a product of primes that is unique up to their order. The property of being prime is called primality. A simple but slow method of checking the primality of a given number n, called trial division, tests whether n is a multiple of any integer between 2 and \sqrt. Faster algorithms include the Miller–Rabin primality test, which is fast but has a small chance of error, and the AKS primality test, which always pr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Heptagonal Numbers
A heptagonal number is a figurate number that is constructed by combining heptagons with ascending size. The ''n''-th heptagonal number is given by the formula :H_n=\frac. The first few heptagonal numbers are: : 0, 1, 7, 18, 34, 55, 81, 112, 148, 189, 235, 286, 342, 403, 469, 540, 616, 697, 783, 874, 970, 1071, 1177, 1288, 1404, 1525, 1651, 1782, … Parity The parity of heptagonal numbers follows the pattern odd-odd-even-even. Like square numbers, the digital root in base 10 of a heptagonal number can only be 1, 4, 7 or 9. Five times a heptagonal number, plus 1 equals a triangular number. Additional properties * The heptagonal numbers have several notable formulas: :H_=H_m+H_n+5mn :H_=H_m+H_n-5mn+3n :H_m-H_n=\frac :40H_n+9=(10n-3)^2 Sum of reciprocals A formula for the sum of the reciprocals of the heptagonal numbers is given by: : \begin\sum_^\infty \frac &= \frac+\frac\ln(5)+\frac\ln\left(\frac\sqrt\right)+\frac\ln\left(\frac\sqrt\right)\\ &=\frac13\left(\frac+\fra ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
