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210 (number)
210 (two hundred [and] ten) is the natural number following 209 (number), 209 and preceding 211 (number), 211. In mathematics 210 is a composite number, an abundant number, Harshad number, and the product of the first four prime numbers (2 (number), 2, 3 (number), 3, 5 (number), 5, and 7 (number), 7), and thus a primorial. It is also the least common multiple of these four prime numbers. It is the sum of eight consecutive prime numbers (13 + 17 + 19 + 23 + 29 + 31 + 37 + 41 = 210).Wells, D. (1987). ''The Penguin Dictionary of Curious and Interesting Numbers'' (p. 143). London: Penguin Group. It is a triangular number (following 190 (number), 190 and preceding 231 (number), 231), a pentagonal number (following 176 (number), 176 and preceding 247 (number), 247), and the second smallest to be both triangular and pentagonal (the third is 40755). It is also an idoneal number, a pentatope number, a pronic number, and an untouchable number. 210 is also the third polygonal number, 71-g ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
1 (number)
1 (one, unit, unity) is a number representing a single or the only entity. 1 is also a numerical digit and represents a single unit of counting or measurement. For example, a line segment of ''unit length'' is a line segment of length 1. In conventions of sign where zero is considered neither positive nor negative, 1 is the first and smallest positive integer. It is also sometimes considered the first of the infinite sequence of natural numbers, followed by 2, although by other definitions 1 is the second natural number, following 0. The fundamental mathematical property of 1 is to be a multiplicative identity, meaning that any number multiplied by 1 equals the same number. Most if not all properties of 1 can be deduced from this. In advanced mathematics, a multiplicative identity is often denoted 1, even if it is not a number. 1 is by convention not considered a prime number; this was not universally accepted until the mid-20th century. Additionally, 1 is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
211 (number)
211 (two hundred ndeleven) is the natural number following 210 and preceding 212. It is also a prime number. In mathematics 211 is an odd number. 211 is a primorial prime, sum of three consecutive primes (67 + 71 + 73), Chen prime, centered decagonal prime, and self prime. 211 is the smallest prime separated by 10 or more from the nearest primes (199 and 223). It is thus a balanced prime and an ''isolated prime''. 211 is a repdigit in base 14 (111). Multiplying its digits, it is still a prime (2), and adding its digits, it is square (4). Rearranging its digits, 211 becomes 121, which also is a square. Adding any two of its digits will be prime (2 or 3). 211 is a super-prime. In science and technology 2-1-1 is special abbreviated telephone number reserved in Canada and the United States as an easy-to-remember three-digit telephone number. It is meant to provide quick information and referrals to health and human service organizations for both services from charities and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pronic Number
A pronic number is a number that is the product of two consecutive integers, that is, a number of the form n(n+1).. The study of these numbers dates back to Aristotle. They are also called oblong numbers, heteromecic numbers,. or rectangular numbers; however, the term "rectangular number" has also been applied to the composite numbers. The first few pronic numbers are: : 0, 2, 6, 12, 20, 30, 42, 56, 72, 90, 110, 132, 156, 182, 210, 240, 272, 306, 342, 380, 420, 462 … . Letting P_n denote the pronic number n(n+1), we have P_ = P_. Therefore, in discussing pronic numbers, we may assume that n\geq 0 without loss of generality, a convention that is adopted in the following sections. As figurate numbers The pronic numbers were studied as figurate numbers alongside the triangular numbers and square numbers in Aristotle's ''Metaphysics'', and their discovery has been attributed much earlier to the Pythagoreans.. As a kind of figurate number, the pronic numbers are somet ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pentatope Number
A pentatope number is a number in the fifth cell of any row of Pascal's triangle starting with the 5-term row , either from left to right or from right to left. The first few numbers of this kind are: : 1, 5, 15, 35, 70, 126, 210, 330, 495, 715, 1001, 1365 Pentatope numbers belong to the class of figurate numbers, which can be represented as regular, discrete geometric patterns. Formula The formula for the th pentatope number is represented by the 4th rising factorial of divided by the factorial of 4: :P_n = \frac = \frac . The pentatope numbers can also be represented as binomial coefficients: :P_n = \binom , which is the number of distinct quadruples that can be selected from objects, and it is read aloud as " plus three choose four". Properties Two of every three pentatope numbers are also pentagonal numbers. To be precise, the th pentatope number is always the th pentagonal number and the th pentatope number is always the th pentagonal number. The th pent ... [...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] |
247 (number)
247 (two hundred ndforty-seven) is the natural number following 246 and preceding 248. Additionally, 247 is: * a semiprime. * a brilliant number (the product of two primes with the same number of digits). * a pentagonal number. * palindromic in base 18 (DD18). * a Harshad number In mathematics, a harshad number (or Niven number) in a given number base is an integer that is divisible by the sum of its digits when written in that base. Harshad numbers in base are also known as -harshad (or -Niven) numbers. Harshad numbers ... in bases 10, 14, 19, 20, 27, 39, 40, 58, 77, 79, 115, 118, 229 and 235. * the smallest number which can be expressed as the difference between two integers that contain together all the decimal digits 0–9. i.e. 247 = 50123 - 49876. References {{Integers, 2 Integers ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
176 (number)
176 (one hundred ndseventy-six) is the natural number following 175 and preceding 177. In mathematics 176 is an even number and an abundant number. It is an odious number, a self number, a semiperfect number, and a practical number. 176 is a cake number, a happy number, a pentagonal number, and an octagonal number. 15 can be partitioned in 176 ways. The Higman–Sims group can be constructed as a doubly transitive permutation group acting on a geometry containing 176 points, and it is also the symmetry group of the largest possible set of equiangular lines in 22 dimensions, which contains 176 lines. In astronomy * 176 Iduna is a large main belt asteroid with a composition similar to that of the largest main belt asteroid, 1 Ceres * Gliese 176 is a red dwarf star in the constellation of Taurus * Gliese 176 b is a super-Earth exoplanet in the constellation of Taurus. This planet orbits close to its parent star Gliese 176 In the Bible * Minuscule 176 (in the Gregor ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pentagonal Number
A pentagonal number is a figurate number that extends the concept of triangular and square numbers to the pentagon, but, unlike the first two, the patterns involved in the construction of pentagonal numbers are not rotationally symmetrical. The ''n''th pentagonal number ''pn'' is the number of ''distinct'' dots in a pattern of dots consisting of the ''outlines'' of regular pentagons with sides up to n dots, when the pentagons are overlaid so that they share one vertex. For instance, the third one is formed from outlines comprising 1, 5 and 10 dots, but the 1, and 3 of the 5, coincide with 3 of the 10 – leaving 12 distinct dots, 10 in the form of a pentagon, and 2 inside. ''p''n is given by the formula: :p_n = =\binom+3\binom for ''n'' ≥ 1. The first few pentagonal numbers are: 1, 5, 12, 22, 35, 51, 70, 92, 117, 145, 176, 210, 247, 287, 330, 376, 425, 477, 532, 590, 651, 715, 782, 852, 925, 1001, 1080, 1162, 1247, 1335, 1426, 1520, 1617, 1717, 1820, 1926, 20 ... [...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] |
190 (number)
190 (one hundred ndninety) is the natural number following 189 and preceding 191. In mathematics 190 is a triangular number, a hexagonal number, and a centered nonagonal number, the fourth figurate number (after 1, 28, and 91) with that combination of properties. It is also a truncated square pyramid number. Integers from 191 to 199 ;191 : 191 is a prime number. ;192 : 192 = 26 × 3 is a 3-smooth number, the smallest number with 14 divisors. ;193 : 193 is a prime number. ;194 : 194 = 2 × 97 is a Markov number, the smallest number written as the sum of three squares in five ways, and the number of irreducible representations of the Monster group. ;195 : 195 = 3 × 5 × 13 is the smallest number expressed as a sum of distinct squares in 16 different ways. ;196 :196 = 22 × 72 is a square number. ;197 :197 is a prime number and a Schröder–Hipparchus number. ;198 :198 = 2 × 32 × 11 is the smallest number written as the sum of four squares in ten ways :No integer factorial eve ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Triangular Number
A triangular number or triangle number counts objects arranged in an equilateral triangle. Triangular numbers are a type of figurate number, other examples being square numbers and cube numbers. The th triangular number is the number of dots in the triangular arrangement with dots on each side, and is equal to the sum of the natural numbers from 1 to . The sequence of triangular numbers, starting with the 0th triangular number, is (This sequence is included in the On-Line Encyclopedia of Integer Sequences .) Formula The triangular numbers are given by the following explicit formulas: T_n= \sum_^n k = 1+2+3+ \dotsb +n = \frac = , where \textstyle is a binomial coefficient. It represents the number of distinct pairs that can be selected from objects, and it is read aloud as " plus one choose two". The first equation can be illustrated using a visual proof. For every triangular number T_n, imagine a "half-square" arrangement of objects corresponding to the triangular numb ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Primorial
In mathematics, and more particularly in number theory, primorial, denoted by "#", is a function from natural numbers to natural numbers similar to the factorial function, but rather than successively multiplying positive integers, the function only multiplies prime numbers. The name "primorial", coined by Harvey Dubner, draws an analogy to ''primes'' similar to the way the name "factorial" relates to ''factors''. Definition for prime numbers For the th prime number , the primorial is defined as the product of the first primes: :p_n\# = \prod_^n p_k, where is the th prime number. For instance, signifies the product of the first 5 primes: :p_5\# = 2 \times 3 \times 5 \times 7 \times 11 = 2310. The first five primorials are: : 2, 6, 30, 210, 2310 . The sequence also includes as empty product. Asymptotically, primorials grow according to: :p_n\# = e^, where is Little O notation. Definition for natural numbers In general, for a positive integer , its pri ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
