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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 ''k'' can be defined formally in terms of the sum-of-divisors function σ1 or the aliquot sum function ''s'' in the following way: : ''s''0 = ''k'' : ''s''n = ''s''(''s''''n''−1) = σ1(''s''''n''−1) − ''s''''n''−1 if ''s''''n''−1 > 0 : ''s''n = 0 if ''s''''n''−1 = 0 ---> (if we add this condition, then the terms after 0 are all 0, and all aliquot sequences would be infinite sequence, and we can conjecture that all aliquot sequences are convergent, the limit of these sequences are usually 0 or 6) and ''s''(0) is undefined. For example, the aliquot sequence of 10 is 10, 8, 7, 1, 0 because: :σ1(10) − 10 = 5 + 2 + 1 = 8, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and mathematical analysis, analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of mathematical object, abstract objects and the use of pure reason to proof (mathematics), prove them. These objects consist of either abstraction (mathematics), abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of inference rule, deductive rules to already established results. These results include previously proved theorems, axioms ... [...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 in the Euclidean space whose lattice points are -tuples of integers. The two-dimensional integer lattice is also called the square lattice, or grid ..., 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. See also * The year AD 124 or 124 BC * 124th (other) * List o ... [...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 (still a conjecture) or Fermat's Last Theorem (a conjecture until proven in 1995 by Andrew Wiles), have shaped much of mathematical history as new areas of mathematics are developed in order to prove them. Important examples Fermat's Last Theorem In number theory, Fermat's Last Theorem (sometimes called Fermat's conjecture, especially in older texts) states that no three positive integers a, ''b'', and ''c'' can satisfy the equation ''a^n + b^n = c^n'' for any integer value of ''n'' greater than two. This theorem was first conjectured by Pierre de Fermat in 1637 in the margin of a copy of '' Arithmetica'', where he claimed that he had a proof that was too large to fit in the margin. The first successful proof was released in 1994 by Andrew Wiles, and formally published in 1995, after 358 years of effort by ... [...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 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. In other fields *"290" was the shipyard number of the ''CSS Alabama'' See also the year 290. Integers from 291 to 299 291 291 = 3·97, a semiprime, floor(3^14/2^14) . 292 292 = 22·73, noncototient, untouchable number. The continued fraction representation of \pi is ; 7, 15, 1, 292, 1, 1, 1, 2... the convergent obtained by truncating before the surprisingly large term 292 yields the excellent rational approximation 355/113 to \pi, repdigit in base 8 (444). 2 ... [...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 sec ... [...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 a 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 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 cycle. In other fields In the Christian calendar, there are 276 days from the Annunciation on March 25 to Christmas on December 25, a number considered significant by some authors. See also *The years 276 and 276 BC __NOTOC__ Ye ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
248 (number)
248 (two hundred ndforty-eight) is the natural number following 247 and preceding 249. In mathematics 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 E8 has dimension 248. In religion *The number 248 is the Gematria value for the Hebrew letters Ramach (Resh Mem and Het), traditionally depicted as the number of organs in the human body, and the number of positive commandments in the Torah. It is also the number of words in the Jewish Shema Prayer, inclusive of בָּרוּךְ שֵׁם כְּבוֹד מַלְכוּתוֹ לְעוֹלָם וָעֶד (Ba-ruch sheim k'vod mal-chu-to l'o-lam va-ed) in response to the first verse, and the repetition of יְהֹוָ֥ה ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
246 (number)
246 (two hundred ndforty-six) is the natural number following 245 and preceding 247. In mathematics 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 with seven black and seven white beads. In other fields *+246 is the code for international direct dial phone calls to British Indian Ocean Territory (Diego Garcia). * +1246, is the area code assigned to Barbados. *List of highways numbered 246. *2-4-6, a Whyte notation classification of steam locomotive A steam locomotive is a locomotive that provides the force ... [...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 is to find a subsequence of a given sequence in which the subsequence's elements are in sorted order, lowest to highest, and in which the subsequence is as long as possible. This subse .... 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 middle ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
210 (number)
210 (two hundred ndten) is the natural number following 209 and preceding 211. In mathematics 210 is a composite number, an abundant number, Harshad number, and the product of the first four prime numbers ( 2, 3, 5, and 7), and thus a primorial. It is also the least common multiple of these four prime numbers. It is the sum of eight consecutive prime numbers (13 + 17 + 19 + 23 + 29 + 31 + 37 + 41 = 210).Wells, D. (1987). ''The Penguin Dictionary of Curious and Interesting Numbers'' (p. 143). London: Penguin Group. It is a 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 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, preceding 418. It is the first primorial number greater than 2 which is not adjacent to 2 primes (211 is prime, but 209 is not). It ... [...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, 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 semigroups of order four up to isomorphism and anti-isomorphism. In science There are 206 bones in the typical adult human body.. See also * The Year 206 AD * Cessna 206, a single engine light aircraft * Bell 206, a light helicopter * The Peugeot 206, a French supermini automobile * US Area code 206 Area code 206 is a North American telephone area code in the U.S. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
