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Octagonal Number
An octagonal number is a figurate number that represents an octagon. The octagonal number for ''n'' is given by the formula 3''n''2 - 2''n'', with ''n'' > 0. The first few octagonal numbers are : 1, 8, 21, 40, 65, 96, 133, 176, 225, 280, 341, 408, 481, 560, 645, 736, 833, 936 Octagonal numbers can be formed by placing triangular numbers on the four sides of a square. To put it algebraically, the ''n''-th octagonal number is x_n=n^2 + 4\sum_^ k = 3n^2-2n. The octagonal number for ''n'' can also be calculated by adding the square of ''n'' to twice the (''n - 1'')th pronic number. Octagonal numbers consistently alternate parity. Octagonal numbers are occasionally referred to as "star numbers," though that term is more commonly used to refer to centered dodecagonal numbers. Applications in combinatorics x_n is the number of partitions of 6n-5 into 1,2 or 3s. For example: there are x_2=8 such partitions for 2*6-5=7: : ,1,1,1,1,1,1 ,1,1,1,1,2 ,1,1,1,3 ,1,1,2,2 ,1,2,3 , ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Figurate Number
The term figurate number is used by different writers for members of different sets of numbers, generalizing from triangular numbers to different shapes (polygonal numbers) and different dimensions (polyhedral numbers). The term can mean * polygonal number * a number represented as a discrete -dimensional regular geometry, geometric pattern of -dimensional Ball (mathematics), balls such as a polygonal number (for ) or a polyhedral number (for ). * a member of the subset of the sets above containing only triangular numbers, pyramidal numbers, and their analogs in other dimensions. Terminology Some kinds of figurate number were discussed in the 16th and 17th centuries under the name "figural number". In historical works about Greek mathematics the preferred term used to be ''figured number''. In a use going back to Jacob Bernoulli's Ars Conjectandi, the term ''figurate number'' is used for triangular numbers made up of successive integers, tetrahedral numbers made up of successi ... [...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] |
Sums Of Reciprocals
In mathematics and especially number theory, the sum of reciprocals generally is computed for the reciprocals of some or all of the positive integers (counting numbers)—that is, it is generally the sum of unit fractions. If infinitely many numbers have their reciprocals summed, generally the terms are given in a certain sequence and the first ''n'' of them are summed, then one more is included to give the sum of the first ''n''+1 of them, etc. If only finitely many numbers are included, the key issue is usually to find a simple expression for the value of the sum, or to require the sum to be less than a certain value, or to determine whether the sum is ever an integer. For an infinite series of reciprocals, the issues are twofold: First, does the sequence of sums diverge—that is, does it eventually exceed any given number—or does it converge, meaning there is some number that it gets arbitrarily close to without ever exceeding it? (A set of positive integers is said to ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Star Number
A star number is a centered figurate number, a centered hexagram (six-pointed star), such as the Star of David, or the board Chinese checkers is played on. The ''n''th star number is given by the formula ''Sn'' = 6''n''(''n'' − 1) + 1. The first 43 star numbers are 1, 13, 37, 73, 121, 181, 253, 337, 433, 541, 661, 793, 937, 1093, 1261, 1441, 1633, 1837, 2053, 2281, 2521, 2773, 3037, 3313, 3601, 3901, 4213, 4537, 4873, 5221, 5581, 5953, 6337, 6733, 7141, 7561, 7993, 8437, 8893, 9361, 9841, 10333, 10837 The digital root of a star number is always 1 or 4, and progresses in the sequence 1, 4, 1. The last two digits of a star number in base 10 are always 01, 13, 21, 33, 37, 41, 53, 61, 73, 81, or 93. Unique among the star numbers is 35113, since its prime factors (i.e., 13, 37 and 73) are also consecutive star numbers. Relationships to other kinds of numbers Geometrically, the ''n''th star number is made up of a central point and 12 copies of the (''n''−1)th triangular n ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Parity (mathematics)
In mathematics, parity is the property of an integer of whether it is even or odd. An integer is even if it is a multiple of two, and odd if it is not.. For example, −4, 0, 82 are even because \begin -2 \cdot 2 &= -4 \\ 0 \cdot 2 &= 0 \\ 41 \cdot 2 &= 82 \end By contrast, −3, 5, 7, 21 are odd numbers. The above definition of parity applies only to integer numbers, hence it cannot be applied to numbers like 1/2 or 4.201. See the section "Higher mathematics" below for some extensions of the notion of parity to a larger class of "numbers" or in other more general settings. Even and odd numbers have opposite parities, e.g., 22 (even number) and 13 (odd number) have opposite parities. In particular, the parity of zero is even. Any two consecutive integers have opposite parity. A number (i.e., integer) expressed in the decimal numeral system is even or odd according to whether its last digit is even or odd. That is, if the last digit is 1, 3, 5, 7, or 9, then it is odd; otherwis ... [...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] |
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] |
280 (number)
280 (two hundred ndeighty) is the natural number after 279 and before 281. In mathematics The denominator of the eighth harmonic number, 280 is an octagonal number. 280 is the smallest octagonal number that is a half of another octagonal number. There are 280 plane trees with ten nodes. As a consequence of this, 18 people around a round table can shake hands with each other in non-crossing ways, in 280 different ways (this includes rotations). In geography *List of highways numbered 280 See also the year 280. Integers from 281 to 289 281 282 282 = 2·3·47, sphenic number, number of planar partitions of 9 283 283 prime, twin prime with 281, strictly non-palindromic number, 4283 - 3283 is prime 284 284 and 220 form the first pair of amicable numbers, as the divisors of 284 add up to 220 and vice versa. 285 285 = 3·5·19, sphenic number, square pyramidal number, Harshad number, repdigit in base 7 (555), vertically symmetric number , also in ''Star Trek'', the total nu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
133 (number)
133 (one hundred ndthirty-three) is the natural number following 132 and preceding 134. In mathematics 133 is an ''n'' whose divisors (excluding ''n'' itself) added up divide φ(''n''). It is an octagonal number and a happy number. 133 is a repdigit in base 11 (111) and base 18 (77), whilst in base 20 it is a cyclic number formed from the reciprocal of the number three. 133 is a semiprime: a product of two prime numbers, namely 7 and 19. Since those prime factors are Gaussian primes, this means that 133 is a Blum integer. 133 is the number of compositions of 13 into distinct parts. In the military * Douglas C-133 Cargomaster was a United States cargo aircraft built between 1956 and 1961 * is a heavy landing craft which launched in 1972 * was a United States Navy ''Mission Buenaventura''-class fleet oilers during World War II * was a United States Navy during World War II * was a United States Navy during World War II * was a United States Navy ''General G. O. Squ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Octagon
In geometry, an octagon (from the Greek ὀκτάγωνον ''oktágōnon'', "eight angles") is an eight-sided polygon or 8-gon. A '' regular octagon'' has Schläfli symbol and can also be constructed as a quasiregular truncated square, t, which alternates two types of edges. A truncated octagon, t is a hexadecagon, . A 3D analog of the octagon can be the rhombicuboctahedron with the triangular faces on it like the replaced edges, if one considers the octagon to be a truncated square. Properties of the general octagon The sum of all the internal angles of any octagon is 1080°. As with all polygons, the external angles total 360°. If squares are constructed all internally or all externally on the sides of an octagon, then the midpoints of the segments connecting the centers of opposite squares form a quadrilateral that is both equidiagonal and orthodiagonal (that is, whose diagonals are equal in length and at right angles to each other).Dao Thanh Oai (2015), "Equilatera ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
96 (number)
96 (ninety-six) is the natural number following 95 and preceding 97. It is a number that appears the same when turned upside down. In mathematics 96 is: * an octagonal number. * a refactorable number. * an untouchable number. * a semiperfect number since it is a multiple of 6. * an abundant number since the sum of its proper divisors is greater than 96. * the fourth Granville number and the second non-perfect Granville number. The next Granville number is 126, the previous being 24. * the sum of Euler's totient function φ(''x'') over the first seventeen integers. * strobogrammatic in bases 10 (9610), 11 (8811) and 95 (1195). * palindromic in bases 11 (8811), 15 (6615), 23 (4423), 31 (3331), 47 (2247) and 95 (1195). * an Erdős–Woods number, since it is possible to find sequences of 96 consecutive integers such that each inner member shares a factor with either the first or the last member. * divisible by the number of prime numbers (24) below 96. Every integer greater ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
65 (number)
65 (sixty-five) is the natural number following 64 and preceding 66. In mathematics Sixty-five is the 23rd semiprime and the 3rd of the form (5.q). It is an octagonal number. It is also a Cullen number. Given 65, the Mertens function returns 0. This number is the magic constant of a 5x5 normal magic square: \begin 17 & 24 & 1 & 8 & 15 \\ 23 & 5 & 7 & 14 & 16 \\ 4 & 6 & 13 & 20 & 22 \\ 10 & 12 & 19 & 21 & 3 \\ 11 & 18 & 25 & 2 & 9 \end. This number is also the magic constant of n-Queens Problem for n = 5. 65 is the smallest integer that can be expressed as a sum of two distinct positive squares in two ways, 65 = 82 + 12 = 72 + 42. It appears in the Padovan sequence, preceded by the terms 28, 37, 49 (it is the sum of the first two of these). There are only 65 known Euler's idoneal numbers. 65 is a Stirling number of the second kind, the number of ways of dividing a set of six objects into four non-empty subsets. 65 = 15 + 24 + 33 + 42 + 51. 65 is the length of the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
