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112 (number)
112 (one hundred [and] twelve) is the natural number following 111 (number), 111 and preceding 113 (number), 113. Mathematics 112 is an abundant number, a heptagonal number, and a Harshad number. 112 is the number of connected graphs on 6 unlabeled nodes. If an equilateral triangle has sides of length 112, then it contains an interior point at integer distances 57, 65, and 73 from its vertices. This is the smallest possible side length of an equilateral triangle that contains a point at integer distances from the vertices.Wells, D. ''The Penguin Dictionary of Curious and Interesting Numbers'' London: Penguin Group. (1987), page 119 See also * 112 (other) References Integers {{Num-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Natural Number
In mathematics, the natural numbers are the numbers 0, 1, 2, 3, and so on, possibly excluding 0. Some start counting with 0, defining the natural numbers as the non-negative integers , while others start with 1, defining them as the positive integers Some authors acknowledge both definitions whenever convenient. Sometimes, the whole numbers are the natural numbers as well as zero. In other cases, the ''whole numbers'' refer to all of the integers, including negative integers. The counting numbers are another term for the natural numbers, particularly in primary education, and are ambiguous as well although typically start at 1. The natural numbers are used for counting things, like "there are ''six'' coins on the table", in which case they are called ''cardinal numbers''. They are also used to put things in order, like "this is the ''third'' largest city in the country", which are called ''ordinal numbers''. Natural numbers are also used as labels, like Number (sports), jersey ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
111 (number)
111 (one hundred [and] eleven) is the natural number following 110 (number), 110 and preceding 112 (number), 112. In mathematics 111 is the fourth non-trivial nonagonal number, and the seventh perfect totient number. 111 is furthermore the ninth number such that its Euler totient \varphi(n) of 72 (number), 72 is equal to the totient value of its Divisor function, sum-of-divisors: :\varphi(111) = \varphi(\sigma(111)). Two other of its multiples (333 (number), 333 and 555 (number), 555) also have the same property (with totients of 216 (number), 216 and 288 (number), 288, respectively). Magic squares The smallest magic square using only 1 and prime numbers has a magic constant of 111: Also, a six-by-six magic square using the numbers 1 to 36 also has a magic constant of 111: (The square has this magic constant because 1 + 2 + 3 + ... + 34 + 35 + 36 = 666 (number), 666, and 666 / 6 = 111). On the other hand, 111 lies between 110 (number), 110 and 112 (number), ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
113 (number)
113 (one hundred ndthirteen) is the natural number following 112 and preceding 114. Mathematics * 113 is the 30th prime number (following 109 and preceding 127), so it can only be divided by one and itself. 113 is a Sophie Germain prime, an emirp, an isolated prime, a Chen prime and a Proth prime as it is a prime number of the form 7\times 2^+1. 113 is also an Eisenstein prime with no imaginary part and real part of the form 3n - 1. In decimal, this prime is a primeval number and a permutable prime with 131 and 311. *113 is a highly cototient number and a centered square number. *113 is the denominator of 355/113, an accurate approximation to . Other uses *113 is also the atomic number of nihonium. * A113 is a Pixar recurring inside joke or Easter Egg, e.g.: (WALL-E ''WALL-E'' (stylized with an interpunct as ''WALL·E'') is a 2008 American animated Romance film, romantic science fiction film produced by Pixar Animation Studios for Walt Disney Pictures. Th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Abundant Number
In number theory, an abundant number or excessive number is a positive integer for which the sum of its proper divisors is greater than the number. The integer 12 is the first abundant number. Its proper divisors are 1, 2, 3, 4 and 6 for a total of 16. The amount by which the sum exceeds the number is the abundance. The number 12 has an abundance of 4, for example. Definition An ''abundant number'' is a natural number for which the Divisor function, sum of divisors satisfies , or, equivalently, the sum of proper divisors (or aliquot sum) satisfies . The ''abundance'' of a natural number is the integer (equivalently, ). Examples The first 28 abundant numbers are: :12, 18, 20, 24, 30, 36, 40, 42, 48, 54, 56, 60, 66, 70, 72, 78, 80, 84, 88, 90, 96, 100, 102, 104, 108, 112, 114, 120, ... . For example, the proper divisors of 24 are 1, 2, 3, 4, 6, 8, and 12, whose sum is 36. Because 36 is greater than 24, the number 24 is abundant. Its abundance is 36 − 24&nb ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Heptagonal Number
In mathematics, 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)\\ &=\fra ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Harshad Number
In mathematics, a harshad number (or Niven number) in a given radix, number base is an integer that is divisible by the digit sum, sum of its digits when written in that base. Harshad numbers in base are also known as -harshad (or -Niven) numbers. Because being a Harshad number is determined based on the base the number is expressed in, a number can be a Harshad number many times over. So-called Trans-Harshad numbers are Harshad numbers in every base. Harshad numbers were defined by D. R. Kaprekar, a mathematician from India. The word "harshad" comes from the Sanskrit ' (joy) + ' (give), meaning joy-giver. The term "Niven number" arose from a paper delivered by Ivan M. Niven at a conference on number theory in 1977. Definition Stated mathematically, let be a positive integer with digits when written in base , and let the digits be a_i (i = 0, 1, \ldots, m-1). (It follows that a_i must be either zero or a positive integer up to .) can be expressed as :X=\sum_^ a_i n^i. is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Connected Graph
In mathematics and computer science, connectivity is one of the basic concepts of graph theory: it asks for the minimum number of elements (nodes or edges) that need to be removed to separate the remaining nodes into two or more isolated subgraphs. It is closely related to the theory of network flow problems. The connectivity of a graph is an important measure of its resilience as a network. Connected vertices and graphs In an undirected graph , two vertices and are called connected if contains a path from to . Otherwise, they are called disconnected. If the two vertices are additionally connected by a path of length (that is, they are the endpoints of a single edge), the vertices are called adjacent. A graph is said to be connected if every pair of vertices in the graph is connected. This means that there is a path between every pair of vertices. An undirected graph that is not connected is called disconnected. An undirected graph is therefore disconnected if there e ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Equilateral Triangle
An equilateral triangle is a triangle in which all three sides have the same length, and all three angles are equal. Because of these properties, the equilateral triangle is a regular polygon, occasionally known as the regular triangle. It is the special case of an isosceles triangle by modern definition, creating more special properties. The equilateral triangle can be found in various tilings, and in polyhedrons such as the deltahedron and antiprism. It appears in real life in popular culture, architecture, and the study of stereochemistry resembling the molecular known as the trigonal planar molecular geometry. Properties An equilateral triangle is a triangle that has three equal sides. It is a special case of an isosceles triangle in the modern definition, stating that an isosceles triangle is defined at least as having two equal sides. Based on the modern definition, this leads to an equilateral triangle in which one of the three sides may be considered its base. Th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
The Penguin Dictionary Of Curious And Interesting Numbers
''The Penguin Dictionary of Curious and Interesting Numbers'' is a reference book for recreational mathematics and elementary number theory written by David Wells. The first edition was published in paperback by Penguin Books in 1986 in the UK, and a revised edition appeared in 1997 (). Contents The entries are arranged in increasing order of magnitude, with the exception of the first entry on −1 and ''i''. The book includes some irrational numbers below 10 but concentrates on integers, and has an entry for every integer up to 42. The final entry is for Graham's number. In addition to the dictionary itself, the book includes a list of mathematicians in chronological sequence (all born before 1890), a short glossary, and a brief bibliography. The back of the book contains eight short tables "for the benefit of readers who cannot wait to look for their own patterns and properties", including lists of polygonal numbers, Fibonacci numbers, prime numbers, factorials, decimal r ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
112 (other) , synthetic chemical element with atomic number 112
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112 may refer to: *112 (number), the natural number following 111 and preceding 113 *112 (band), an American R&B quartet from Atlanta, Georgia ** ''112'' (album), album from the band of the same name *112 (emergency telephone number), the standard emergency phone number in the European Union and on GSM cellphones *112 BC, a year *AD 112, a year of the Julian calendar *Copernicium, an element with atomic number 112 *112 (MBTA bus) *112 (New Jersey bus) *KFM 112M aircraft engine *Thai Criminal Code section 112, see Lèse majesté in Thailand *112 Iphigenia, a main-belt asteroid See also * 1/12 (other) * 11/2 (other) * I12 (other) *Copernicium Copernicium is a synthetic chemical element; it has symbol Cn and atomic number 112. Its known isotopes are extremely radioactive, and have only been created in a laboratory. The most stable known isotope, copernicium-285, has a half-life of ap ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
