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232 (number)
232 (two hundred ndthirty-two) is the natural number following 231 and preceding 233. In mathematics 232 is both a central polygonal number and a cake number. It is both a decagonal number and a centered 11-gonal number. It is also a refactorable number, a Motzkin sum, an idoneal number, a Riordan number and a noncototient. 232 is a telephone number: in a system of seven telephone users, there are 232 different ways of pairing up some of the users. There are also exactly 232 different eight-vertex connected indifference graphs, and 232 bracelets A bracelet is an article of jewellery that is worn around the wrist. Bracelets may serve different uses, such as being worn as an ornament. When worn as ornaments, bracelets may have a supportive function to hold other items of decoration, suc ... with eight beads of one color and seven of another. Because this number has the form , it follows that there are exactly 232 different functions from a set of four elements to a prope ... [...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] |
231 (number)
231 (two hundred ndthirty-one) is the natural number following 230 and preceding 232. In mathematics Two hundred ndthirty-one 231 = 3·7·11, sphenic number, triangular number, doubly triangular number, hexagonal number, octahedral number, centered octahedral number, the number of integer partitions of 16, Mertens function returns 0, and is the number of cubic inches in a U.S. liquid gallon. In other fields 231 is: * The year A.D. 231 or 231 BC. * +231 is the country code for Liberia. * '' Pacific 231'', an orchestral work by French composer Arthur Honegger. * '' Pacific 231'', a 1949 short film directed by Jean Mitry Jean-René Pierre Goetgheluck Le Rouge Tillard des Acres de Presfontaines, whose pseudonym was Jean Mitry (; 7 November 1904 – 18 January 1988), was a French film theorist, critic and filmmaker, a co-founder of France's first film society, and, .... * List of highways numbered 231 References Integers {{Num-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
233 (number)
233 (two hundred ndthirty-three) is the natural number following 232 and preceding 234. In mathematics *233 is a prime number, *233 is a Sophie Germain prime, a Pillai prime, and a Ramanujan prime. *It is a Fibonacci number, one of the Fibonacci primes. *There are exactly 233 maximal planar graphs with ten vertices, and 233 connected topological spaces with four points. In other fields * +233 is the telephone country code for Ghana Ghana (; tw, Gaana, ee, Gana), officially the Republic of Ghana, is a country in West Africa. It abuts the Gulf of Guinea and the Atlantic Ocean to the south, sharing borders with Ivory Coast in the west, Burkina Faso in the north, and To .... * 233 Celsius is the temperature at which paper burns. References Integers {{Num-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lazy Caterer's Sequence
The lazy caterer's sequence, more formally known as the central polygonal numbers, describes the maximum number of pieces of a disk (a pancake or pizza is usually used to describe the situation) that can be made with a given number of straight cuts. For example, three cuts across a pancake will produce six pieces if the cuts all meet at a common point inside the circle, but up to seven if they do not. This problem can be formalized mathematically as one of counting the cells in an arrangement of lines; for generalizations to higher dimensions, ''see'' arrangement of hyperplanes. The analogue of this sequence in three dimensions is the cake number. Formula and sequence The maximum number ''p'' of pieces that can be created with a given number of cuts , where , is given by the formula : p = \frac. Using binomial coefficients, the formula can be expressed as :p = 1 + \dbinom = \dbinom+\dbinom+\dbinom. Simply put, each number equals a triangular number plus 1. As the third col ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cake Number
In mathematics, the cake number, denoted by ''Cn'', is the maximum of the number of regions into which a 3-dimensional cube can be partitioned by exactly ''n'' planes. The cake number is so-called because one may imagine each partition of the cube by a plane as a slice made by a knife through a cube-shaped cake. It is the 3D analogue of the lazy caterer's sequence. The values of ''Cn'' for increasing are given by General formula If ''n''! denotes the factorial, and we denote the binomial coefficients by : = \frac , and we assume that ''n'' planes are available to partition the cube, then the ''n''-th cake number is: : C_n = + + + = \tfrac\left(n^3 + 5n + 6\right) = \tfrac\left(n+1) (n (n-1) + 6\right). Properties The only cake number which is prime is 2, since it requires \left(n+1) (n (n-1) + 6\right) to have prime factorisation 2 \cdot 3 \cdot p where p is some prime. This is impossible for n > 2 as we know \left(n (n-1) + 6\right) must be even, so it must be e ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Decagonal Number
A decagonal number is a figurate number that extends the concept of triangular and square numbers to the decagon (a ten-sided polygon). However, unlike the triangular and square numbers, the patterns involved in the construction of decagonal numbers are not rotationally symmetrical. Specifically, the ''n''th decagonal numbers counts the number of dots in a pattern of ''n'' nested decagons, all sharing a common corner, where the ''i''th decagon in the pattern has sides made of ''i'' dots spaced one unit apart from each other. The ''n''-th decagonal number is given by the following formula : D_n = 4n^2 - 3n. The first few decagonal numbers are: : 0, 1, 10, 27, 52, 85, 126, 175, 232, 297, 370, 451, 540, 637, 742, 855, 976, 1105, 1242, 1387, 1540, 1701, 1870, 2047, 2232, 2425, 2626, 2835, 3052, 3277, 3510, 3751, 4000, 4257, 4522, 4795, 5076, 5365, 5662, 5967, 6280, 6601, 6930, 7267, 7612, 7965, 8326 The ''n''th decagonal number can also be calculated by adding the square of '' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Refactorable Number
A refactorable number or tau number is an integer ''n'' that is divisible by the count of its divisors, or to put it algebraically, ''n'' is such that \tau(n)\mid n. The first few refactorable numbers are listed in as : 1, 2, 8, 9, 12, 18, 24, 36, 40, 56, 60, 72, 80, 84, 88, 96, 104, 108, 128, 132, 136, 152, 156, 180, 184, 204, 225, 228, 232, 240, 248, 252, 276, 288, 296, ... For example, 18 has 6 divisors (1 and 18, 2 and 9, 3 and 6) and is divisible by 6. There are infinitely many refactorable numbers. Properties Cooper and Kennedy proved that refactorable numbers have natural density zero. Zelinsky proved that no three consecutive integers can all be refactorable. Colton proved that no refactorable number is perfect. The equation \gcd(n,x) = \tau(n) has solutions only if n is a refactorable number, where \gcd is the greatest common divisor function. Let T(x) be the number of refactorable numbers which are at most x. The problem of determining an a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Idoneal Number
In mathematics, Leonhard Euler, Euler's idoneal numbers (also called suitable numbers or convenient numbers) are the positive integers ''D'' such that any integer expressible in only one way as ''x''2 ± ''Dy''2 (where ''x''2 is relatively prime to ''Dy''2) is a prime power or twice a prime power. In particular, a number that has two distinct representations as a sum of two squares is Euler's factorization method, composite. Every idoneal number generates a set containing infinitely many primes and missing infinitely many other primes. Definition A positive integer ''n'' is idoneal if and only if it cannot be written as ''ab'' + ''bc'' + ''ac'' for distinct positive integers ''a, b'', and ''c''. It is sufficient to consider the set ; if all these numbers are of the form , , or ''2''s for some integer s, where is a prime, then is idoneal. Conjecturally complete listing The 65 idoneal numbers found by Leonhard Euler and Carl Friedrich Gauss and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
A005043
A, or a, is the first letter and the first vowel of the Latin alphabet, used in the modern English alphabet, the alphabets of other western European languages and others worldwide. Its name in English is ''a'' (pronounced ), plural ''aes''. It is similar in shape to the Ancient Greek letter alpha, from which it derives. The uppercase version consists of the two slanting sides of a triangle, crossed in the middle by a horizontal bar. The lowercase version can be written in two forms: the double-storey a and single-storey ɑ. The latter is commonly used in handwriting and fonts based on it, especially fonts intended to be read by children, and is also found in italic type. In English grammar, " a", and its variant " an", are indefinite articles. History The earliest certain ancestor of "A" is aleph (also written 'aleph), the first letter of the Phoenician alphabet, which consisted entirely of consonants (for that reason, it is also called an abjad to distinguish it fro ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Noncototient
In mathematics, a noncototient is a positive integer ''n'' that cannot be expressed as the difference between a positive integer ''m'' and the number of coprime integers below it. That is, ''m'' − φ(''m'') = ''n'', where φ stands for Euler's totient function, has no solution for ''m''. The ''cototient'' of ''n'' is defined as ''n'' − φ(''n''), so a noncototient is a number that is never a cototient. It is conjectured that all noncototients are even. This follows from a modified form of the slightly stronger version of the Goldbach conjecture: if the even number ''n'' can be represented as a sum of two distinct primes ''p'' and ''q,'' then :pq - \varphi(pq) = pq - (p-1)(q-1) = p+q-1 = n-1. \, It is expected that every even number larger than 6 is a sum of two distinct primes, so probably no odd number larger than 5 is a noncototient. The remaining odd numbers are covered by the observations 1=2-\phi(2), 3 = 9 - \phi(9) and 5 = 25 - ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Telephone Number (mathematics)
In mathematics, the telephone numbers or the involution numbers form a sequence of integers that count the ways people can be connected by person-to-person telephone calls. These numbers also describe the number of matchings (the Hosoya index) of a complete graph on vertices, the number of permutations on elements that are involutions, the sum of absolute values of coefficients of the Hermite polynomials, the number of standard Young tableaux with cells, and the sum of the degrees of the irreducible representations of the symmetric group. Involution numbers were first studied in 1800 by Heinrich August Rothe, who gave a recurrence equation by which they may be calculated, giving the values (starting from ) Applications John Riordan provides the following explanation for these numbers: suppose that people subscribe to a telephone service that can connect any two of them by a call, but cannot make a single call connecting more than two people. How many different patterns ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Indifference Graph
In graph theory, a branch of mathematics, an indifference graph is an undirected graph constructed by assigning a real number to each vertex and connecting two vertices by an edge when their numbers are within one unit of each other.. Indifference graphs are also the intersection graphs of sets of unit intervals, or of properly nested intervals (intervals none of which contains any other one). Based on these two types of interval representations, these graphs are also called unit interval graphs or proper interval graphs; they form a subclass of the interval graphs. Equivalent characterizations The finite indifference graphs may be equivalently characterized as *The intersection graphs of unit intervals, *The intersection graphs of sets of intervals no two of which are nested (one containing the other),. *The claw-free interval graphs, *The graphs that do not have an induced subgraph isomorphic to a claw ''K''1,3, net (a triangle with a degree-one vertex adjacent to each of the tr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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