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252 (number)
252 (two hundred ndfifty-two) is the natural number following 251 and preceding 253. In mathematics 252 is: *the central binomial coefficient \tbinom, the largest one divisible by all coefficients in the previous line *\tau(3), where \tau is the Ramanujan tau function. *\sigma_3(6), where \sigma_3 is the function that sums the cubes of the divisors of its argument: :1^3+2^3+3^3+6^3=(1^3+2^3)(1^3+3^3)=252. *a practical number, *a refactorable number, *a hexagonal pyramidal number. *a member of the Mian-Chowla sequence. There are 252 points on the surface of a cuboctahedron of radius five in the face-centered cubic lattice, 252 ways of writing the number 4 as a sum of six squares of integers, 252 ways of choosing four squares from a 4×4 chessboard up to reflections and rotations, and 252 ways of placing three pieces on a Connect Four Connect Four (also known as Connect 4, Four Up, Plot Four, Find Four, Captain's Mistress, Four in a Row, Drop Four, and Gravitrips in ... [...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 number, cardinal numbers'', and numbers used for ordering are called ''Ordinal number, 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 Number (sports), jersey numbers). Some definitions, including the standard ISO/IEC 80000, 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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
251 (number)
251 (two hundred ndfifty-one) is the natural number between 250 and 252. It is also a prime number. In mathematics 251 is: *a Sophie Germain prime. *the sum of three consecutive primes (79 + 83 + 89) and seven consecutive primes (23 + 29 + 31 + 37 + 41 + 43 + 47). *a Chen prime. *an Eisenstein prime with no imaginary part. *a de Polignac number, meaning that it is odd and cannot be formed by adding a power of two to a prime number. *the smallest number that can be formed in more than one way by summing three positive cubes:251 = 2^3 + 3^3 + 6^3 = 1^3 + 5^3 + 5^3. Every 5 × 5 matrix has exactly 251 square submatrices. In science *The average atomic mass and most stable isotope of Californium, which has a half life Half-life (symbol ) is the time required for a quantity (of substance) to reduce to half of its initial value. The term is commonly used in nuclear physics to describe how quickly unstable atoms undergo radioactive decay or how long stable at ... ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
253 (number)
253 (two hundred ndfifty-three) is the natural number following 252 and preceding 254. In mathematics 253 is: *a semiprime since it is the product of 2 primes. *a triangular number. *a star number. *a centered heptagonal number. *a centered nonagonal number. *a Blum integer In mathematics, a natural number ''n'' is a Blum integer if is a semiprime for which ''p'' and ''q'' are distinct prime numbers congruent to 3 mod 4.Joe Hurd, Blum Integers (1997), retrieved 17 Jan, 2011 from http://www.gilith.com/research/tal .... *a member of the 13-aliquot tree. References {{Integers, 2 Integers ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Central Binomial Coefficient
In mathematics the ''n''th central binomial coefficient is the particular binomial coefficient : = \frac = \prod\limits_^\frac \textn \geq 0. They are called central since they show up exactly in the middle of the even-numbered rows in Pascal's triangle. The first few central binomial coefficients starting at ''n'' = 0 are: :, , , , , , 924, 3432, 12870, 48620, ...; Properties The central binomial coefficients represent the number of combinations of a set where there are an equal number of two types of objects. For example, n=2 represents ''AABB, ABAB, ABBA, BAAB, BABA, BBAA''. They also represent the number of combinations of ''A'' and ''B'' where there are never more ''B'' 's than ''A'' 's. For example, n=2 represents ''AAAA, AAAB, AABA, AABB, ABAA, ABAB''. The number of factors of ''2'' in \binom is equal to the number of ones in the binary representation of ''n'', so ''1'' is the only odd central binomial coefficient. Generating function The ordinary generating fun ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ramanujan Tau Function
The Ramanujan tau function, studied by , is the function \tau : \mathbb \rarr\mathbb defined by the following identity: :\sum_\tau(n)q^n=q\prod_\left(1-q^n\right)^ = q\phi(q)^ = \eta(z)^=\Delta(z), where with , \phi is the Euler function, is the Dedekind eta function, and the function is a holomorphic cusp form of weight 12 and level 1, known as the discriminant modular form (some authors, notably Apostol, write \Delta/(2\pi)^ instead of \Delta). It appears in connection to an "error term" involved in counting the number of ways of expressing an integer as a sum of 24 squares. A formula due to Ian G. Macdonald was given in . Values The first few values of the tau function are given in the following table : Ramanujan's conjectures observed, but did not prove, the following three properties of : * if (meaning that is a multiplicative function) * for prime and . * for all primes . The first two properties were proved by and the third one, called the Ramanujan conjec ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Divisor Function
In mathematics, and specifically in number theory, a divisor function is an arithmetic function related to the divisors of an integer. When referred to as ''the'' divisor function, it counts the ''number of divisors of an integer'' (including 1 and the number itself). It appears in a number of remarkable identities, including relationships on the Riemann zeta function and the Eisenstein series of modular forms. Divisor functions were studied by Ramanujan, who gave a number of important Modular arithmetic, congruences and identity (mathematics), identities; these are treated separately in the article Ramanujan's sum. A related function is the divisor summatory function, which, as the name implies, is a sum over the divisor function. Definition The sum of positive divisors function σ''z''(''n''), for a real or complex number ''z'', is defined as the summation, sum of the ''z''th Exponentiation, powers of the positive divisors of ''n''. It can be expressed in Summation#Capital ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Practical Number
In number theory, a practical number or panarithmic number is a positive integer n such that all smaller positive integers can be represented as sums of distinct divisors of n. For example, 12 is a practical number because all the numbers from 1 to 11 can be expressed as sums of its divisors 1, 2, 3, 4, and 6: as well as these divisors themselves, we have 5 = 3 + 2, 7 = 6 + 1, 8 = 6 + 2, 9 = 6 + 3, 10 = 6 + 3 + 1, and 11 = 6 + 3 + 2. The sequence of practical numbers begins Practical numbers were used by Fibonacci in his Liber Abaci (1202) in connection with the problem of representing rational numbers as Egyptian fractions. Fibonacci does not formally define practical numbers, but he gives a table of Egyptian fraction expansions for fractions with practical denominators.. The name "practical number" is due to . He noted that "the subdivisions of money, weights, and measures involve numbers like 4, 12, 16, 20 and 28 which are usually supposed to be so inconvenient as to dese ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Refactorable Number
A refactorable number or tau number is an integer ''n'' that is divisible by the count of its divisors, or to put it algebraically, ''n'' is such that \tau(n)\mid n. The first few refactorable numbers are listed in as : 1, 2, 8, 9, 12, 18, 24, 36, 40, 56, 60, 72, 80, 84, 88, 96, 104, 108, 128, 132, 136, 152, 156, 180, 184, 204, 225, 228, 232, 240, 248, 252, 276, 288, 296, ... For example, 18 has 6 divisors (1 and 18, 2 and 9, 3 and 6) and is divisible by 6. There are infinitely many refactorable numbers. Properties Cooper and Kennedy proved that refactorable numbers have natural density zero. Zelinsky proved that no three consecutive integers can all be refactorable. Colton proved that no refactorable number is perfect. The equation \gcd(n,x) = \tau(n) has solutions only if n is a refactorable number, where \gcd is the greatest common divisor function. Let T(x) be the number of refactorable numbers which are at most x. The problem of determining an a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hexagonal Pyramidal Number
A pyramidal number is a figurate number that represents a pyramid with a polygonal base and a given number of triangular sides. A pyramidal number is the number of points in a pyramid where each layer of the pyramid is an -sided polygon of points. The term often refers to square pyramidal numbers, which have a square base with four sides, but it can also refer to pyramids with three or more sides. The numbers of points in the base (and in parallel layers to the base) are given by polygonal numbers of the given number of sides, while the numbers of points in each triangular side is given by a triangular number. It is possible to extend the pyramidal numbers to higher dimensions. Formula The formula for the th -gonal pyramidal number is :P_n^r= \frac, where , . This formula can be factored: :P_n^r=\frac=\left(\frac\right)\left(\frac\right)=T_n \cdot \frac, where is the th triangular number. Sequences The first few triangular pyramidal numbers (equivalently, tetrahedral numb ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mian–Chowla Sequence
In mathematics, the Mian–Chowla sequence is an integer sequence defined recursion, recursively in the following way. The sequence starts with :a_1 = 1. Then for n>1, a_n is the smallest integer such that every pairwise sum :a_i + a_j is distinct, for all i and j less than or equal to n. Properties Initially, with a_1, there is only one pairwise sum, 1 + 1 = 2. The next term in the sequence, a_2, is 2 since the pairwise sums then are 2, 3 and 4, i.e., they are distinct. Then, a_3 can't be 3 because there would be the non-distinct pairwise sums 1 + 3 = 2 + 2 = 4. We find then that a_3 = 4, with the pairwise sums being 2, 3, 4, 5, 6 and 8. The sequence thus begins :1 (number), 1, 2 (number), 2, 4 (number), 4, 8 (number), 8, 13 (number), 13, 21 (number), 21, 31 (number), 31, 45 (number), 45, 66 (number), 66, 81 (number), 81, 97 (number), 97, 123 (number), 123, 148 (number), 148, 182 (number), 182, 204 (number), 204,252 (number), 252, 290 (number), 290, 361, 401, 475, ... . ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cuboctahedron
A cuboctahedron is a polyhedron with 8 triangular faces and 6 square faces. A cuboctahedron has 12 identical vertices, with 2 triangles and 2 squares meeting at each, and 24 identical edges, each separating a triangle from a square. As such, it is a quasiregular polyhedron, i.e. an Archimedean solid that is not only vertex-transitive but also edge-transitive. It is radially equilateral. Its dual polyhedron is the rhombic dodecahedron. The cuboctahedron was probably known to Plato: Heron's ''Definitiones'' quotes Archimedes as saying that Plato knew of a solid made of 8 triangles and 6 squares. Synonyms *''Vector Equilibrium'' (Buckminster Fuller) because its center-to-vertex radius equals its edge length (it has radial equilateral symmetry). Fuller also called a cuboctahedron built of rigid struts and flexible vertices a ''jitterbug''; this object can be progressively transformed into an icosahedron, octahedron, and tetrahedron by folding along the diagonals of its square sid ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
FCC Close Packing
In geometry, close-packing of equal spheres is a dense arrangement of congruent spheres in an infinite, regular arrangement (or lattice). Carl Friedrich Gauss proved that the highest average density – that is, the greatest fraction of space occupied by spheres – that can be achieved by a lattice packing is :\frac \approx 0.74048. The same packing density can also be achieved by alternate stackings of the same close-packed planes of spheres, including structures that are aperiodic in the stacking direction. The Kepler conjecture states that this is the highest density that can be achieved by any arrangement of spheres, either regular or irregular. This conjecture was proven by T. C. Hales. Highest density is known only for 1, 2, 3, 8, and 24 dimensions. Many crystal structures are based on a close-packing of a single kind of atom, or a close-packing of large ions with smaller ions filling the spaces between them. The cubic and hexagonal arrangements are very close to one ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
