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Aliquot Sequence
In mathematics, an aliquot sequence is a sequence of positive integers in which each term is the sum of the proper divisors of the previous term. If the sequence reaches the number 1, it ends, since the sum of the proper divisors of 1 is 0. Definition and overview The aliquot sequence starting with a positive integer can be defined formally in terms of the sum-of-divisors function or the aliquot sum function in the following way: \begin s_0 &= k \\ pts_n &= s(s_) = \sigma_1(s_) - s_ \quad \text \quad s_ > 0 \\ pts_n &= 0 \quad \text \quad s_ = 0 \\ pts(0) &= \text \end If the condition is added, then the terms after 0 are all 0, and all aliquot sequences would be infinite, and we can conjecture that all aliquot sequences are convergent, the limit of these sequences are usually 0 or 6. For example, the aliquot sequence of 10 is because: \begin \sigma_1(10) -10 &= 5 + 2 + 1 = 8, \\ pt\sigma_1(8) - 8 &= 4 + 2 + 1 = 7, \\ pt\sigma_1(7) - 7 &= 1, \\ pt\sigma_1(1) - 1 &= 0. ... [...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] |
124 (number)
124 (one hundred ndtwenty-four) is the natural number following 123 and preceding 125. In mathematics 124 is an untouchable number, meaning that it is not the sum of proper divisors of any positive number. It is a stella octangula number, the number of spheres packed in the shape of a stellated octahedron. It is also an icosahedral number. There are 124 different polygons of length 12 formed by edges of the integer lattice In mathematics, the -dimensional integer lattice (or cubic lattice), denoted , is the lattice (group), lattice in the Euclidean space whose lattice points are tuple, -tuples of integers. The two-dimensional integer lattice is also called the s ..., counting two polygons as the same only when one is a translated copy of the other. 124 is a perfectly partitioned number, meaning that it divides the number of partitions of 124. It is the first number to do so after 1, 2, and 3. References Integers {{Num-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Conjecture
In mathematics, a conjecture is a conclusion or a proposition that is proffered on a tentative basis without proof. Some conjectures, such as the Riemann hypothesis or Fermat's conjecture (now a theorem, proven in 1995 by Andrew Wiles), have shaped much of mathematical history as new areas of mathematics are developed in order to prove them. Resolution of conjectures Proof Formal mathematics is based on ''provable'' truth. In mathematics, any number of cases supporting a universally quantified conjecture, no matter how large, is insufficient for establishing the conjecture's veracity, since a single counterexample could immediately bring down the conjecture. Mathematical journals sometimes publish the minor results of research teams having extended the search for a counterexample farther than previously done. For instance, the Collatz conjecture, which concerns whether or not certain sequences of integers terminate, has been tested for all integers up to 1.2 × 101 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
290 (number)
290 (two hundred ndninety) is the natural number following 289 and preceding 291. In mathematics The product of three primes, 290 is a sphenic number, and the sum of four consecutive primes (67 + 71 + 73 + 79). The sum of the squares of the divisors of 17 is 290. Not only is it a 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 nontotie ... and a noncototient, it is also an untouchable number. 290 is the 16th member of the Mian–Chowla sequence; it can not be obtained as the sum of any two previous terms in the sequence. See also the Bhargava–Hanke 290 theorem. References {{DEFAULTSORT:290 (Number) Integers ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
288 (number)
288 (two hundred ndeighty-eight) is the natural number following 287 and preceding 289. Because 288 = 2 · 12 · 12, it may also be called "two gross" or "two dozen dozen". In mathematics Factorization properties Because its prime factorization 288 = 2^5\cdot 3^2 contains only the first two prime numbers 2 and 3, 288 is a 3-smooth number. This factorization also makes it a highly powerful number, a number with a record-setting value of the product of the exponents in its factorization. Among the highly abundant numbers, numbers with record-setting sums of divisors, it is one of only 13 such numbers with an odd divisor sum. Both 288 and are powerful numbers, numbers in which all exponents of the prime factorization are larger than one. This property is closely connected to being highly abundant with an odd divisor sum: all sufficiently large highly abundant numbers have an odd prime factor with exponent one, causing their divisor sum to be even. 288 and 289 form only the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
276 (number)
276 (two hundred ndseventy-six) is the natural number following 275 and preceding 277. In mathematics 276 is the sum of 3 consecutive fifth powers (276 = 15 + 25 + 35). As a figurate number it is the 23rd triangular number, a hexagonal number, and a centered pentagonal number, the third number after 1 and 6 to have this combination of properties. 276 is the first triangular number that can be arrived at in three ways by adding pairs of triangular numbers together. This sequence, dubbed 'Triple Triangle-Pair Numbers' is the sequence of integers: 276, 406, 666, ... 276 is the size of the largest set of equiangular lines in 23 dimensions. The maximal set of such lines, derived from the Leech lattice, provides the highest dimension in which the "Gerzon bound" of \binom is known to be attained; its symmetry group is the third Conway group, Co3. 276 is the smallest number for which it is not known if the corresponding aliquot sequence either terminates or ends in a repeating cyc ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
248 (number)
248 (two hundred ndforty-eight) is the natural number following 247 and preceding 249. Additionally, 248 is: *a nontotient. *a refactorable number. *an untouchable number. *palindromic in bases 13 (16113), 30 (8830), 61 (4461) and 123 (22123). *a Harshad number in bases 3, 4, 6, 7, 9, 11, 13 (and 18 other bases). *part of the 43-aliquot tree. The aliquot sequence starting at 248 is: 248, 232, 218, 112, 136, 134, 70, 74, 40, 50, 43, 1, 0. The exceptional Lie group In mathematics, a Lie group (pronounced ) is a group (mathematics), group that is also a differentiable manifold, such that group multiplication and taking inverses are both differentiable. A manifold is a space that locally resembles Eucli ... E8 has dimension 248. References Integers {{Num-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
246 (number)
246 (two hundred ndforty-six) is the natural number following 245 and preceding 247. Additionally, 246 is: *an untouchable number. *palindromic in bases 5 (14415), 9 (3039), 40 (6640), 81 (3381), 122 (22122) and 245 (11245). *a Harshad number in bases 2, 3, 6, 7, 9, 11 (and 15 other bases). *the smallest number N for which it is known that there is an infinite number of prime gaps no larger than N. Also: *The aliquot sequence starting at 246 is: 246, 258, 270, 450, 759, 393, 135, 105, 87, 33, 15, 9, 4, 3, 1, 0. *There are exactly 246 different rooted plane trees with eight nodes, and 246 different necklaces A necklace is an article of jewellery that is worn around the neck. Necklaces may have been one of the earliest types of adornment worn by humans. They often serve Ceremony, ceremonial, Religion, religious, magic (illusion), magical, or Funerar ... with seven black and seven white beads. References Integers {{Num-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
238 (number)
238 (two hundred ndthirty-eight) is the natural number following 237 and preceding 239. In mathematics 238 is an untouchable number. There are 238 2-vertex-connected graphs on five labeled vertices, and 238 order-5 polydiamonds (polyiamonds that can partitioned into 5 diamonds). Out of the 720 permutations of six elements, exactly 238 of them have a unique longest increasing subsequence In computer science, the longest increasing subsequence problem aims to find a subsequence of a given sequence in which the subsequence's elements are sorted in an ascending order and in which the subsequence is as long as possible. This subsequenc .... There are 238 compact and paracompact hyperbolic groups of ranks 3 through 10. References Integers {{Num-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
216 (number)
216 (two hundred ndsixteen) is the natural number following 215 and preceding 217. It is a cube, and is often called Plato's number, although it is not certain that this is the number intended by Plato. In mathematics 216 is the cube of 6, and the sum of three cubes:216=6^3=3^3+4^3+5^3. It is the smallest cube that can be represented as a sum of three positive cubes, making it the first nontrivial example for Euler's sum of powers conjecture. It is, moreover, the smallest number that can be represented as a sum of any number of distinct positive cubes in more than one way. It is a highly powerful number: the product 3\times 3 of the exponents in its prime factorization 216 = 2^3\times 3^3 is larger than the product of exponents of any smaller number. Because there is no way to express it as the sum of the proper divisors of any other integer, it is an untouchable number. Although it is not a semiprime, the three closest numbers on either side of it are, making it the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
210 (number)
210 (two hundred ndten) is the natural number following 209 and preceding 211. Mathematics 210 is an abundant number, and Harshad number. It is the product of the first four prime numbers ( 2, 3, 5, and 7), and thus a primorial, where it is the least common multiple of these four prime numbers. 210 is the first primorial number greater than 2 which is not adjacent to 2 primes (211 is prime, but 209 is not). It is the sum of eight consecutive prime numbers, between 13 and the thirteenth prime number: Wells, D. (1987). ''The Penguin Dictionary of Curious and Interesting Numbers'' (p. 143). London: Penguin Group. It is the 20th triangular number (following 190 and preceding 231), a pentagonal number (following 176 and preceding 247), and the second smallest to be both triangular and pentagonal (the first is 1; 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 71-gonal number, pre ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
206 (number)
206 (two hundred ndsix) is the natural number following 205 and preceding 207. In mathematics 206 is both a nontotient and a noncototient. 206 is an untouchable number. It is the lowest positive integer (when written in English as "two hundred and six") to employ all of the vowels once only, not including Y. The other numbers sharing this property are 230, 250, 260, 602, 640, 5000, 8000, 9000, 26,000, 80,000 and 90,000. 206 and 207 form the second pair of consecutive numbers (after 14 and 15) whose sums of divisors are equal. There are exactly 206 different linear forests on five labeled nodes, and exactly 206 regular semigroup In mathematics, a regular semigroup is a semigroup ''S'' in which every element is regular, i.e., for each element ''a'' in ''S'' there exists an element ''x'' in ''S'' such that . Regular semigroups are one of the most-studied classes of semigroup ...s of order four up to isomorphism and anti-isomorphism. References Integers {{Num-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
