|
Octagonal Number
In mathematics, an octagonal number is a figurate number. The ''n''th octagonal number ''o''''n'' is the number of dots in a pattern of dots consisting of the outlines of regular octagons with sides up to ''n'' dots, when the octagons are overlaid so that they share one vertex (geometry), vertex. 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 (number), 1, 8 (number), 8, 21 (number), 21, 40 (number), 40, 65 (number), 65, 96 (number), 96, 133 (number), 133, 176 (number), 176, 225 (number), 225, 280 (number), 280, 341, 408, 481, 560, 645, 736, 833, 936 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 (mathematics), parity. Octagonal numbers are occasionally referred to as "star numbers", though that term is more commonly used to refer to centered dodecagonal numbers. Appl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Centered Octagonal Number
A centered octagonal number is a centered number, centered figurate number that represents an octagon with a dot in the center and all other dots surrounding the center dot in successive octagonal layers.. The centered octagonal numbers are the same as the odd number, odd square numbers. Thus, the ''n''th odd square number and ''t''th centered octagonal number is given by the formula :O_n=(2n-1)^2 = 4n^2-4n+1 , (2t+1)^2=4t^2+4t+1. The first few centered octagonal numbers are :1 (number), 1, 9 (number), 9, 25 (number), 25, 49 (number), 49, 81 (number), 81, 121 (number), 121, 169 (number), 169, 225, 289, 361, 441, 529, 625, 729, 841, 961, 1089 (number), 1089, 1225 Calculating Ramanujan's tau function on a centered octagonal number yields an odd number, whereas for any other number the function yields an even number. O_n is the number of 2x2 matrices with elements from 0 to n that their determinant is twice their Permanent (mathematics), permanent. See also * Octagonal number ... [...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 In mathematics, an octagonal number is a figurate number. The ''n''th octagonal number ''o'n'' is the number of dots in a pattern of dots consisting of the outlines of regular octagons with sides up to ''n'' dots, when the octagons are overlai .... 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. References External links Number Facts and Trivia: 176The Number 176The Positive Integer 176 Number Gossip: 176 {{D ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sums Of Reciprocals
In mathematics and especially number theory, the sum of reciprocals (or sum of inverses) 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 posit ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Integer Partition
In number theory and combinatorics, a partition of a non-negative integer , also called an integer partition, is a way of writing as a summation, sum of positive integers. Two sums that differ only in the order of their summands are considered the same partition. (If order matters, the sum becomes a composition (combinatorics), composition.) For example, can be partitioned in five distinct ways: : : : : : The only partition of zero is the empty sum, having no parts. The order-dependent composition is the same partition as , and the two distinct compositions and represent the same partition as . An individual summand in a partition is called a part. The number of partitions of is given by the Partition function (number theory), partition function . So . The notation means that is a partition of . Partitions can be graphically visualized with Young diagrams or Ferrers diagrams. They occur in a number of branches of mathematics and physics, including the study of symm ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Star Number
In mathematics, 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 45 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, 11353, and 11881. 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 copi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Parity (mathematics)
In mathematics, parity is the Property (mathematics), property of an integer of whether it is even or odd. An integer is even if it is divisible by 2, and odd if it is not.. For example, −4, 0, and 82 are even numbers, while −3, 5, 23, and 69 are odd numbers. The above definition of parity applies only to integer numbers, hence it cannot be applied to numbers with decimals or fractions like 1/2 or 4.6978. 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; otherwise it is even—as ... [...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 60 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, 506, 552, 600, 650, 702, 756, 812, 870, 930, 992, 1056, 1122, 1190, 1260, 1332, 1406, 1482, 1560, 1640, 1722, 1806, 1892, 1980, 2070, 2162, 2256, 2352, 2450, 2550, 2652, 2756, 2862, 2970, 3080, 3192, 3306, 3422, 3540, 3660... . 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 nu ... [...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 tree In botany, a tree is a perennial plant with an elongated stem, or trunk, usually supporting branches and leaves. In some usages, the definition of a tree may be narrower, e.g., including only woody plants with secondary growth, only ...s 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). References {{DEFAULTSORT:280 (Number) Integers ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
225 (number)
225 (two hundred ndtwenty-five) is the natural number following 224 and preceding 226. In mathematics 225 is the smallest number that is a polygonal number in five different ways. It is a square number , an octagonal number, and a squared triangular number . As the square of a double factorial, counts the number of permutations of six items in which all cycles have even length, or the number of permutations in which all cycles have odd length. And as one of the Stirling numbers of the first kind, it counts the number of permutations of six items with exactly three cycles. 225 is a highly composite odd number, meaning that it has more divisors than any smaller odd numbers. After 1 and 9, 225 is the third smallest number ''n'' for which , where ''σ'' is the sum of divisors function and ''φ'' is Euler's totient function. 225 is a refactorable number. 225 is the smallest square number In mathematics, a square number or perfect square is an integer that is the squar ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many areas of mathematics, which include number theory (the study of numbers), algebra (the study of formulas and related structures), geometry (the study of shapes and spaces that contain them), Mathematical analysis, analysis (the study of continuous changes), and set theory (presently used as a foundation for all mathematics). Mathematics involves the description and manipulation of mathematical object, abstract objects that consist of either abstraction (mathematics), abstractions from nature orin modern mathematicspurely abstract entities that are stipulated to have certain properties, called axioms. Mathematics uses pure reason to proof (mathematics), prove properties of objects, a ''proof'' consisting of a succession of applications of in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
96 (number)
96 (ninety-six) is the natural number following 95 (number), 95 and preceding 97 (number), 97. It is a number that Strobogrammatic number, appears the same when rotated by 180 degrees. 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 (number), 126, the previous being 24 (number), 24. * the sum of Euler's totient function φ(''x'') over the first seventeen integers. * Strobogrammatic number, strobogrammatic in bases 10 (9610), 11 (8811) and 95 (1195). * Palindromic number, 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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
