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146 (number)
146 (one hundred ndforty-six) is the natural number following 145 and preceding 147. In mathematics 146 is an octahedral number, the number of spheres that can be packed into in a regular octahedron with six spheres along each edge. For an octahedron with seven spheres along each edge, the number of spheres on the surface of the octahedron is again 146. It is also possible to arrange 146 disks in the plane into an irregular octagon with six disks on each side, making 146 an octo number. There is no integer with exactly 146 coprimes less than it, so 146 is a nontotient. It is also never the difference between an integer and the total of coprimes below it, so it is a noncototient. And it is not the sum of proper divisors of any number, making it an untouchable number. There are 146 connected partially ordered set In mathematics, especially order theory, a partially ordered set (also poset) formalizes and generalizes the intuitive concept of an ordering, sequencing, or arran ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Octahedral Number
In number theory, an octahedral number is a figurate number that represents the number of spheres in an octahedron formed from close-packed spheres. The ''n''th octahedral number O_n can be obtained by the formula:. :O_n=. The first few octahedral numbers are: : 1, 6, 19, 44, 85, 146, 231, 344, 489, 670, 891 . Properties and applications The octahedral numbers have a generating function : \frac = \sum_^ O_n z^n = z +6z^2 + 19z^3 + \cdots . Sir Frederick Pollock conjectured in 1850 that every positive integer is the sum of at most 7 octahedral numbers. This statement, the Pollock octahedral numbers conjecture, has been proven true for all but finitely many numbers. In chemistry, octahedral numbers may be used to describe the numbers of atoms in octahedral clusters; in this context they are called magic numbers.. Relation to other figurate numbers Square pyramids An octahedral packing of spheres may be partitioned into two square pyramids, one upside-down underneath ... [...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 numbers'', and numbers used for ordering are called ''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 jersey numbers). Some definitions, including the standard 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 numbers form a set. Many other number sets are built by succ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
145 (number)
145 (one hundred ndforty-five) is the natural number following 144 and preceding 146. In mathematics * Although composite, 145 is a Fermat pseudoprime to sixteen bases with b < 145. In four of those bases, it is a strong pseudoprime: 1, 12, 17, and 144. * Given 145, the Mertens function returns 0. * 145 is a and a centered square number. * . 145 is the fourth number that is the sum of two different pairs of < ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
147 (number)
147 (one hundred ndforty-seven) is the natural number following 146 and preceding 148. In mathematics 147 is the fourth centered icosahedral number. These are a class of figurate numbers that represent points in the shape of a regular icosahedron or alternatively points in the shape of a cuboctahedron, and are magic numbers for the face-centered cubic lattice. Separately, it is also a magic number for the diamond cubic. It is also the fourth Apéry number a_3, where a_n=\sum_^n\binom^2\binom. There are 147 different ways of representing one as a sum of unit fractions with five terms, allowing repeated fractions, and 147 different self-avoiding polygonal chains of length six using horizontal and vertical segments of the integer lattice. In other fields 147 is the highest possible break in snooker, in the absence of fouls and refereeing errors. In some traditions, there are 147 psalms. However, current Christian and Jewish traditions list a larger number, leading to the s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Octahedral Number
In number theory, an octahedral number is a figurate number that represents the number of spheres in an octahedron formed from close-packed spheres. The ''n''th octahedral number O_n can be obtained by the formula:. :O_n=. The first few octahedral numbers are: : 1, 6, 19, 44, 85, 146, 231, 344, 489, 670, 891 . Properties and applications The octahedral numbers have a generating function : \frac = \sum_^ O_n z^n = z +6z^2 + 19z^3 + \cdots . Sir Frederick Pollock conjectured in 1850 that every positive integer is the sum of at most 7 octahedral numbers. This statement, the Pollock octahedral numbers conjecture, has been proven true for all but finitely many numbers. In chemistry, octahedral numbers may be used to describe the numbers of atoms in octahedral clusters; in this context they are called magic numbers.. Relation to other figurate numbers Square pyramids An octahedral packing of spheres may be partitioned into two square pyramids, one upside-down underneath ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Regular Octahedron
In geometry, an octahedron (plural: octahedra, octahedrons) is a polyhedron with eight faces. The term is most commonly used to refer to the regular octahedron, a Platonic solid composed of eight equilateral triangles, four of which meet at each vertex. A regular octahedron is the dual polyhedron of a cube. It is a rectified tetrahedron. It is a square bipyramid in any of three orthogonal orientations. It is also a triangular antiprism in any of four orientations. An octahedron is the three-dimensional case of the more general concept of a cross polytope. A regular octahedron is a 3-ball in the Manhattan () metric. Regular octahedron Dimensions If the edge length of a regular octahedron is ''a'', the radius of a circumscribed sphere (one that touches the octahedron at all vertices) is :r_u = \frac a \approx 0.707 \cdot a and the radius of an inscribed sphere (tangent to each of the octahedron's faces) is :r_i = \frac a \approx 0.408\cdot a while the midradius, whic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Coprime
In mathematics, two integers and are coprime, relatively prime or mutually prime if the only positive integer that is a divisor of both of them is 1. Consequently, any prime number that divides does not divide , and vice versa. This is equivalent to their greatest common divisor (GCD) being 1. One says also '' is prime to '' or '' is coprime with ''. The numbers 8 and 9 are coprime, despite the fact that neither considered individually is a prime number, since 1 is their only common divisor. On the other hand, 6 and 9 are not coprime, because they are both divisible by 3. The numerator and denominator of a reduced fraction are coprime, by definition. Notation and testing Standard notations for relatively prime integers and are: and . In their 1989 textbook '' Concrete Mathematics'', Ronald Graham, Donald Knuth, and Oren Patashnik proposed that the notation a\perp b be used to indicate that and are relatively prime and that the term "prime" be used instead of coprime ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nontotient
In number theory, a nontotient is a positive integer ''n'' which is not a totient number: it is not in the range of Euler's totient function φ, that is, the equation φ(''x'') = ''n'' has no solution ''x''. In other words, ''n'' is a nontotient if there is no integer ''x'' that has exactly ''n'' coprimes below it. All odd numbers are nontotients, except 1, since it has the solutions ''x'' = 1 and ''x'' = 2. The first few even nontotients are : 14, 26, 34, 38, 50, 62, 68, 74, 76, 86, 90, 94, 98, 114, 118, 122, 124, 134, 142, 146, 152, 154, 158, 170, 174, 182, 186, 188, 194, 202, 206, 214, 218, 230, 234, 236, 242, 244, 246, 248, 254, 258, 266, 274, 278, 284, 286, 290, 298, ... Least ''k'' such that the totient of ''k'' is ''n'' are (0 if no such ''k'' exists) :1, 3, 0, 5, 0, 7, 0, 15, 0, 11, 0, 13, 0, 0, 0, 17, 0, 19, 0, 25, 0, 23, 0, 35, 0, 0, 0, 29, 0, 31, 0, 51, 0, 0, 0, 37, 0, 0, 0, 41, 0, 43, 0, 69, 0, 47, 0, 65, 0, 0, 0, 53, 0, 81, 0, 87, ... [...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] |
Untouchable Number
An untouchable number is a positive integer that cannot be expressed as the sum of all the proper divisors of any positive integer (including the untouchable number itself). That is, these numbers are not in the image of the aliquot sum function. Their study goes back at least to Abu Mansur al-Baghdadi (circa 1000 AD), who observed that both 2 and 5 are untouchable. Examples For example, the number 4 is not untouchable as it is equal to the sum of the proper divisors of 9: 1 + 3 = 4. The number 5 is untouchable as it is not the sum of the proper divisors of any positive integer: 5 = 1 + 4 is the only way to write 5 as the sum of distinct positive integers including 1, but if 4 divides a number, 2 does also, so 1 + 4 cannot be the sum of all of any number's proper divisors (since the list of factors would have to contain both 4 and 2). The first few untouchable numbers are: : 2, 5, 52, 88, 96, 120, 124, 146, 162, 188, 2 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Partially Ordered Set
In mathematics, especially order theory, a partially ordered set (also poset) formalizes and generalizes the intuitive concept of an ordering, sequencing, or arrangement of the elements of a set. A poset consists of a set together with a binary relation indicating that, for certain pairs of elements in the set, one of the elements precedes the other in the ordering. The relation itself is called a "partial order." The word ''partial'' in the names "partial order" and "partially ordered set" is used as an indication that not every pair of elements needs to be comparable. That is, there may be pairs of elements for which neither element precedes the other in the poset. Partial orders thus generalize total orders, in which every pair is comparable. Informal definition A partial order defines a notion of comparison. Two elements ''x'' and ''y'' may stand in any of four mutually exclusive relationships to each other: either ''x'' ''y'', or ''x'' and ''y'' are ''inc ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
146 (other)
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{{Number disambiguation ...
146 may refer to: *146 (number), a natural number * AD 146, a year in the 2nd century AD *146 BC, a year in the 2nd century BC *146 (Antrim Artillery) Corps Engineer Regiment, Royal Engineers See also * List of highways numbered 146 The following highways are numbered 146: Brazil * BR-146 Canada * Prince Edward Island Route 146 Costa Rica * National Route 146 India * National Highway 146 (India) Japan * Japan National Route 146 * Fukuoka Prefectural Route 146 * Nara ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
