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Highly Abundant Number
In mathematics, a highly abundant number is a natural number with the property that the sum of its divisors (including itself) is greater than the sum of the divisors of any smaller natural number. Highly abundant numbers and several similar classes of numbers were first introduced by , and early work on the subject was done by . Alaoglu and Erdős tabulated all highly abundant numbers up to 104, and showed that the number of highly abundant numbers less than any ''N'' is at least proportional to log2 ''N''. Formal definition and examples Formally, a natural number ''n'' is called highly abundant if and only if for all natural numbers ''m'' < ''n'', :\sigma(n) > \sigma(m) where σ denotes the sum-of-divisors function. The first few highly abundant numbers are : 1, 2, 3, 4, 6, 8, 10, 12, 16, 18, 20, 24, 30, 36, 42, 48, 60, ... . For instance, 5 is not highly abundant because σ(5) = 5+1 = 6 is smaller than σ(4) = 4 + 2 + 1 = 7, while 8 is highly abundant b ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Highly Abundant Number Cuisenaire Rods 8
High may refer to: Science and technology * Height * High (atmospheric), a high-pressure area * High (computability), a quality of a Turing degree, in computability theory * High (tectonics), in geology an area where relative tectonic uplift took or takes place * Substance intoxication, also known by the slang description "being high" * Sugar high, a misconception about the supposed psychological effects of sucrose Music Performers * High (musical group), a 1974–1990 Indian rock group * The High, an English rock band formed in 1989 Albums * ''High'' (The Blue Nile album) or the title song, 2004 * ''High'' (Flotsam and Jetsam album), 1997 * ''High'' (New Model Army album) or the title song, 2007 * ''High'' (Royal Headache album) or the title song, 2015 * ''High'' (EP), by Jarryd James, or the title song, 2016 Songs * "High" (Alison Wonderland song), 2018 * "High" (The Chainsmokers song), 2022 * "High" (The Cure song), 1992 * "High" (David Hallyday song), 1988 * ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
20 (number)
20 (twenty; Roman numeral XX) is the natural number following 19 and preceding 21. A group of twenty units may also be referred to as a score. In mathematics *20 is a pronic number. *20 is a tetrahedral number as 1, 4, 10, 20. *20 is the basis for vigesimal number systems. *20 is the third composite number to be the product of a squared prime and a prime, and also the second member of the (''2''2)''q'' family in this form. *20 is the smallest primitive abundant number. *An icosahedron has 20 faces. A dodecahedron has 20 vertices. *20 can be written as the sum of three Fibonacci numbers uniquely, i.e. 20 = 13 + 5 + 2. *20 is the number of moves (quarter or half turns) required to optimally solve a Rubik's Cube in the worst case. (e.g. the newspaper headline "Scores of Typhoon Survivors Flown to Manila")."CBS News"''Scores of Typhoon Survivors Flown to Manila'' (November 2013) In sports * Twenty20 is a form of limited overs cricket where each team plays only 20 overs. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Transactions Of The American Mathematical Society
The ''Transactions of the American Mathematical Society'' is a monthly peer-reviewed scientific journal of mathematics published by the American Mathematical Society. It was established in 1900. As a requirement, all articles must be more than 15 printed pages. See also * ''Bulletin of the American Mathematical Society'' * '' Journal of the American Mathematical Society'' * ''Memoirs of the American Mathematical Society'' * ''Notices of the American Mathematical Society'' * ''Proceedings of the American Mathematical Society'' External links * ''Transactions of the American Mathematical Society''on JSTOR JSTOR (; short for ''Journal Storage'') is a digital library founded in 1995 in New York City. Originally containing digitized back issues of academic journals, it now encompasses books and other primary sources as well as current issues of j ... American Mathematical Society academic journals Mathematics journals Publications established in 1900 {{math-journal-st ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Powerful Number
A powerful number is a positive integer ''m'' such that for every prime number ''p'' dividing ''m'', ''p''2 also divides ''m''. Equivalently, a powerful number is the product of a square and a cube, that is, a number ''m'' of the form ''m'' = ''a''2''b''3, where ''a'' and ''b'' are positive integers. Powerful numbers are also known as squareful, square-full, or 2-full. Paul Erdős and George Szekeres studied such numbers and Solomon W. Golomb named such numbers ''powerful''. The following is a list of all powerful numbers between 1 and 1000: :1, 4, 8, 9, 16, 25, 27, 32, 36, 49, 64, 72, 81, 100, 108, 121, 125, 128, 144, 169, 196, 200, 216, 225, 243, 256, 288, 289, 324, 343, 361, 392, 400, 432, 441, 484, 500, 512, 529, 576, 625, 648, 675, 676, 729, 784, 800, 841, 864, 900, 961, 968, 972, 1000, ... . Equivalence of the two definitions If ''m'' = ''a''2''b''3, then every prime in the prime factorization of ''a'' appears in the prime factorization of ''m'' with an exponent of at ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Abundant Number
In number theory, an abundant number or excessive number is a number for which the sum of its proper divisors is greater than the number. The integer 12 is the first abundant number. Its proper divisors are 1, 2, 3, 4 and 6 for a total of 16. The amount by which the sum exceeds the number is the abundance. The number 12 has an abundance of 4, for example. Definition A number ''n'' for which the ''sum'' ''of'' ''divisors'' ''σ''(''n'') > 2''n'', or, equivalently, the sum of proper divisors (or aliquot sum) ''s''(''n'') > ''n''. Abundance is the value ''σ''(''n'') − ''2n'' (or ''s''(''n'') − ''n''). Examples The first 28 abundant numbers are: :12, 18, 20, 24, 30, 36, 40, 42, 48, 54, 56, 60, 66, 70, 72, 78, 80, 84, 88, 90, 96, 100, 102, 104, 108, 112, 114, 120, ... . For example, the proper divisors of 24 are 1, 2, 3, 4, 6, 8, and 12, whose sum is 36. Because 36 is greater than 24, the number 24 is abundant. Its abundance is 36 − 24 = 12. Prope ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Superabundant Number
In mathematics, a superabundant number (sometimes abbreviated as SA) is a certain kind of natural number. A natural number ''n'' is called superabundant precisely when, for all ''m'' < ''n'' :\frac 6/5. Superabundant numbers were defined by . Unknown to Alaoglu and Erdős, about 30 pages of Ramanujan's 1915 paper "Highly Composite Numbers" were suppressed. Those pages were finally published in The Ramanujan Journal 1 (1997), 119–153. In section 59 of that paper, Ramanujan defines generalized highly composite numbers, which include the superabundant numbers. Properties proved that if ''n'' is superabundant, then there exist a ''k'' and ''a''1, ''a''2, ..., ''a''''k'' such that :n=\prod_^k (p_i)^ where ''p''i is the ''i''-th prime number, and :a_1\geq a_2\geq\dotsb\geq a_k\geq 1. That is, they proved that if ''n'' is superabundant, the prime decomposition of ''n'' has non-increasing exponents (the exponent of a larger prime is never more than that a smaller prime) and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Factorial
In mathematics, the factorial of a non-negative denoted is the product of all positive integers less than or equal The factorial also equals the product of n with the next smaller factorial: \begin n! &= n \times (n-1) \times (n-2) \times (n-3) \times \cdots \times 3 \times 2 \times 1 \\ &= n\times(n-1)!\\ \end For example, 5! = 5\times 4! = 5 \times 4 \times 3 \times 2 \times 1 = 120. The value of 0! is 1, according to the convention for an empty product. Factorials have been discovered in several ancient cultures, notably in Indian mathematics in the canonical works of Jain literature, and by Jewish mystics in the Talmudic book '' Sefer Yetzirah''. The factorial operation is encountered in many areas of mathematics, notably in combinatorics, where its most basic use counts the possible distinct sequences – the permutations – of n distinct objects: there In mathematical analysis, factorials are used in power series for the exponential function an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Counterexample
A counterexample is any exception to a generalization. In logic a counterexample disproves the generalization, and does so rigorously in the fields of mathematics and philosophy. For example, the fact that "John Smith is not a lazy student" is a counterexample to the generalization “students are lazy”, and both a counterexample to, and disproof of, the universal quantification “all students are lazy.” In mathematics, the term "counterexample" is also used (by a slight abuse) to refer to examples which illustrate the necessity of the full hypothesis of a theorem. This is most often done by considering a case where a part of the hypothesis is not satisfied and the conclusion of the theorem does not hold. In mathematics In mathematics, counterexamples are often used to prove the boundaries of possible theorems. By using counterexamples to show that certain conjectures are false, mathematical researchers can then avoid going down blind alleys and learn to modify conjectures t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
60 (number)
60 (sixty) () is the natural number following 59 and preceding 61. Being three times 20, it is called '' threescore'' in older literature ('' kopa'' in Slavic, ''Schock'' in Germanic). In mathematics * 60 is a highly composite number. Because it is the sum of its unitary divisors (excluding itself), it is a unitary perfect number, and it is an abundant number with an abundance of 48. Being ten times a perfect number, it is a semiperfect number. * It is the smallest number divisible by the numbers 1 to 6: there is no smaller number divisible by the numbers 1 to 5. * It is the smallest number with exactly 12 divisors. * It is one of seven integers that have more divisors than any number less than twice itself , one of six that are also lowest common multiple of a consecutive set of integers from 1, and one of six that are divisors of every highly composite number higher than itself. * It is the smallest number that is the sum of two odd primes in six ways.Wells, D. ''The Penguin D ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
48 (number)
48 (forty-eight) is the natural number following 47 and preceding 49. It is one third of a gross, or four dozens. In mathematics Forty-eight is the double factorial of 6, a highly composite number. Like all other multiples of 6, it is a semiperfect number. 48 is the second 17- gonal number. 48 is the smallest number with exactly ten divisors. There are 11 solutions to the equation φ(''x'') = 48, namely 65, 104, 105, 112, 130, 140, 144, 156, 168, 180 and 210. This is more than any integer below 48, making 48 a highly totient number. Since the greatest prime factor of 482 + 1 = 2305 is 461, which is clearly more than twice 48, 48 is a Størmer number. 48 is a Harshad number in base 10. It has 24, 2, 12, and 4 as factors. In science *The atomic number of cadmium. *The number of Ptolemaic constellations. *The number of symmetries of a cube. Astronomy *Messier object M48, a magnitude 5.5 open cluster in the constellation Hydra. *The New General Cataloguebr>objec ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
42 (number)
42 (forty-two) is the natural number that follows 41 (number), 41 and precedes 43 (number), 43. Mathematics Forty-two (42) is a pronic number and an abundant number; its prime factorization (2\times 3\times 7) makes it the second sphenic number and also the second of the form (2\times 3\times r). Additional properties of the number 42 include: * It is the number of isomorphism classes of all simple and oriented directed graphs on 4 vertices. In other words, it is the number of all possible outcomes (up to isomorphism) of a tournament consisting of 4 teams where the game between any pair of teams results in three possible outcomes: the first team wins, the second team wins, or there is a draw. The group stage of the FIFA World cup is a good example. * It is the third primary pseudoperfect number. * It is a Catalan number. Consequently, 42 is the number of noncrossing partitions of a set of five elements, the number of triangulations of a heptagon, the number of rooted ordered bina ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
36 (number)
36 (thirty-six) is the natural number following 35 and preceding 37. In mathematics 36 is both the square of six and a triangular number, making it a square triangular number. It is the smallest square triangular number other than one, and it is also the only triangular number other than one whose square root is also a triangular number. It is also a Harshad number. It is the smallest number ''n'' with exactly eight solutions to the equation \phi(x)=n. It is the smallest number with exactly nine divisors, leading 36 to be a highly composite number. Adding up some subsets of its divisors (e.g., 6, 12, and 18) gives 36; hence, it is a semiperfect number. This number is the sum of the cubes of the first three positive integers and also the product of the squares of the first three positive integers. 36 is the number of degrees in the interior angle of each tip of a regular pentagram. The thirty-six officers problem is a mathematical puzzle with no solution. The number of possi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
