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Abundant Number
In number theory, an abundant number or excessive number is a positive integer 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 An ''abundant number'' is a natural number for which the Divisor function, sum of divisors satisfies , or, equivalently, the sum of proper divisors (or aliquot sum) satisfies . The ''abundance'' of a natural number is the integer (equivalently, ). 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&nb ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Abundant Number Cuisenaire Rods 12
Abundance may refer to: In science and technology * Abundance (economics), the opposite of scarcities * Abundance (ecology), the relative representation of a species in a community * Abundance, the defining characteristic of abundant numbers * Abundance (programming language), a Forth-like computer programming language * In chemistry: ** Abundance (chemistry), when a substance in a reaction is present in high quantities ** Abundance of the chemical elements, a measure of how common elements are *** Natural abundance, the natural prevalence of different isotopes of an element on Earth *** Abundance of elements in Earth's crust In literature * Al-Kawthar ("Abundance"), the 108th sura of the Qur'an * ''Abundance'' (play), a 1990 stage play written by Beth Henley * '' Abundance: The Future Is Better Than You Think'', a 2012 book by Peter Diamandis and Steven Kotler * ''Abundance'' (novel), a 2021 novel by Jakob Guanzon * ''Abundance'' (Klein and Thompson book), 2025 nonfiction b ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cambridge University Press
Cambridge University Press was the university press of the University of Cambridge. Granted a letters patent by King Henry VIII in 1534, it was the oldest university press in the world. Cambridge University Press merged with Cambridge Assessment to form Cambridge University Press and Assessment under Queen Elizabeth II's approval in August 2021. With a global sales presence, publishing hubs, and offices in more than 40 countries, it published over 50,000 titles by authors from over 100 countries. Its publications include more than 420 academic journals, monographs, reference works, school and university textbooks, and English language teaching and learning publications. It also published Bibles, runs a bookshop in Cambridge, sells through Amazon, and has a conference venues business in Cambridge at the Pitt Building and the Sir Geoffrey Cass Sports and Social Centre. It also served as the King's Printer. Cambridge University Press, as part of the University of Cambridge, was a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Journal Of Number Theory
The ''Journal of Number Theory'' (''JNT'') is a monthly peer-reviewed scientific journal covering all aspects of number theory. The journal was established in 1969 by R.P. Bambah, P. Roquette, A. Ross, A. Woods, and H. Zassenhaus (Ohio State University). It is currently published monthly by Elsevier and the editor-in-chief is Dorian Goldfeld (Columbia University). According to the ''Journal Citation Reports'', the journal has a 2022 impact factor The impact factor (IF) or journal impact factor (JIF) of an academic journal is a type of journal ranking. Journals with higher impact factor values are considered more prestigious or important within their field. The Impact Factor of a journa ... of 0.7. David Goss prize The David Goss Prize in Number theory, founded by the Journal of Number Theory, is awarded every two years, to mathematicians under the age of 35 for outstanding contributions to number theory. The prize is dedicated to the memory of David Goss who was the fo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Friendly Number
In number theory, friendly numbers are two or more natural numbers with a common abundancy index, the ratio between the sum of divisors of a number and the number itself. Two numbers with the same "abundancy" form a friendly pair; ''n'' numbers with the same abundancy form a friendly ''n''-tuple. Being mutually friendly is an equivalence relation, and thus induces a partition of the positive naturals into clubs (equivalence classes) of mutually friendly numbers. A number that is not part of any friendly pair is called solitary. The abundancy index of ''n'' is the rational number σ(''n'') / ''n'', in which σ denotes the sum of divisors function. A number ''n'' is a friendly number if there exists ''m'' ≠ ''n'' such that σ(''m'') / ''m'' = σ(''n'') / ''n''. Abundancy is not the same as abundance, which is defined as σ(''n'') − 2''n''. Abundancy may also be expressed as \sigma_(n) where \sigma_k denotes a divisor function with \sigma_(n) equal to the sum of the ''k''-th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics Magazine
''Mathematics Magazine'' is a refereed bimonthly publication of the Mathematical Association of America. Its intended audience is teachers of collegiate mathematics, especially at the junior/senior level, and their students. It is explicitly a journal of mathematics rather than pedagogy. Rather than articles in the terse "theorem-proof" style of research journals, it seeks articles which provide a context for the mathematics they deliver, with examples, applications, illustrations, and historical background. Paid circulation in 2008 was 9,500 and total circulation was 10,000. ''Mathematics Magazine'' is a continuation of ''Mathematics News Letter'' (1926–1934) and ''National Mathematics Magazine'' (1934–1945). Doris Schattschneider became the first female editor of ''Mathematics Magazine'' in 1981. .. The MAA gives the Carl B. Allendoerfer Awards annually "for articles of expository excellence" published in ''Mathematics Magazine''. See also *''American Mathematical Mon ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Introduction To Arithmetic
Nicomachus of Gerasa (; ) was an Ancient Greek Neopythagoreanism, Neopythagorean philosopher from Gerasa, in the Syria (Roman province), Roman province of Syria (now Jerash, Jordan). Like many Pythagoreans, Nicomachus wrote about the mystical properties of numbers, best known for his works ''Introduction to Arithmetic'' and ''Manual of Harmonics'', which are an important resource on Ancient Greek mathematics and Ancient Greek music in the Roman period. Nicomachus' work on arithmetic became a standard text for Neoplatonic education in Late antiquity, with philosophers such as Iamblichus and John Philoponus writing commentaries on it. A Latin paraphrase by Boethius of Nicomachus's works on arithmetic and music became standard textbooks in medieval education. Life Little is known about the life of Nicomachus except that he was a Pythagoreanism, Pythagorean who came from Gerasa. His ''Manual of Harmonics'' was addressed to a lady of noble birth, at whose request Nicomachus wrote the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nicomachus
Nicomachus of Gerasa (; ) was an Ancient Greek Neopythagorean philosopher from Gerasa, in the Roman province of Syria (now Jerash, Jordan). Like many Pythagoreans, Nicomachus wrote about the mystical properties of numbers, best known for his works ''Introduction to Arithmetic'' and ''Manual of Harmonics'', which are an important resource on Ancient Greek mathematics and Ancient Greek music in the Roman period. Nicomachus' work on arithmetic became a standard text for Neoplatonic education in Late antiquity, with philosophers such as Iamblichus and John Philoponus writing commentaries on it. A Latin paraphrase by Boethius of Nicomachus's works on arithmetic and music became standard textbooks in medieval education. Life Little is known about the life of Nicomachus except that he was a Pythagorean who came from Gerasa. His ''Manual of Harmonics'' was addressed to a lady of noble birth, at whose request Nicomachus wrote the book, which suggests that he was a respected schola ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Deficient Number
In number theory, a deficient number or defective number is a positive integer for which the sum of divisors of is less than . Equivalently, it is a number for which the sum of proper divisors (or aliquot sum) is less than . For example, the proper divisors of 8 are , and their sum is less than 8, so 8 is deficient. Denoting by the sum of divisors, the value is called the number's deficiency. In terms of the aliquot sum , the deficiency is . Examples The first few deficient numbers are :1, 2, 3, 4, 5, 7, 8, 9, 10, 11, 13, 14, 15, 16, 17, 19, 21, 22, 23, 25, 26, 27, 29, 31, 32, 33, 34, 35, 37, 38, 39, 41, 43, 44, 45, 46, 47, 49, 50, ... As an example, consider the number 21. Its divisors are 1, 3, 7 and 21, and their sum is 32. Because 32 is less than 42, the number 21 is deficient. Its deficiency is 2 × 21 − 32 = 10. Properties Since the aliquot sums of prime numbers equal 1, all prime numbers are deficient. More generally, all odd numbers with one or two distinct ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Quasiperfect Number
In mathematics, a quasiperfect number is a natural number for which the sum of all its divisors (the sum-of-divisors function \sigma(n)) is equal to 2n + 1. Equivalently, is the sum of its non-trivial divisors (that is, its divisors excluding 1 and ). No quasiperfect numbers have been found so far. The quasiperfect numbers are the abundant numbers of minimal abundance (which is 1). Theorems If a quasiperfect number exists, it must be an odd square number greater than 1035 and have at least seven distinct prime factor A prime number (or a prime) is a natural number greater than 1 that is not a product of two smaller natural numbers. A natural number greater than 1 that is not prime is called a composite number. For example, 5 is prime because the only ways ...s. Related For a perfect number the sum of all its divisors is equal to 2n. For an almost perfect number the sum of all its divisors is equal to 2n - 1. Numbers whose sum of factors equals 2n + 2 are known to ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Weird Number
In number theory, a weird number is a natural number that is abundant but not semiperfect. In other words, the sum of the proper divisors (divisors including 1 but not itself) of the number is greater than the number, but no subset of those divisors sums to the number itself. Examples The smallest weird number is 70. Its proper divisors are 1, 2, 5, 7, 10, 14, and 35; these sum to 74, but no subset of these sums to 70. The number 12, for example, is abundant but ''not'' weird, because the proper divisors of 12 are 1, 2, 3, 4, and 6, which sum to 16; but 2 + 4 + 6 = 12. The first several weird numbers are : 70, 836, 4030, 5830, 7192, 7912, 9272, 10430, 10570, 10792, 10990, 11410, 11690, 12110, 12530, 12670, 13370, 13510, 13790, 13930, 14770, ... . Properties Infinitely many weird numbers exist. For example, 70''p'' is weird for all primes ''p'' ≥ 149. In fact, the set of weird numbers has positive asymptotic density. It is not known if ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Semiperfect Number
In number theory, a semiperfect number or pseudoperfect number is a natural number ''n'' that is equal to the sum of all or some of its proper divisors. A semiperfect number that is equal to the sum of all its proper divisors is a perfect number. The first few semiperfect numbers are: 6, 12, 18, 20, 24, 28, 30, 36, 40, ... Properties * Every multiple of a semiperfect number is semiperfect. A semiperfect number that is not divisible by any smaller semiperfect number is called ''primitive''. * Every number of the form 2''m''''p'' for a natural number ''m'' and an odd prime number ''p'' such that ''p'' < 2''m''+1 is also semiperfect. ** In particular, every number of the form 2''m''(2''m''+1 − 1) is semiperfect, and indeed perfect if 2''m''+1 − 1 is a . * The sm ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Superabundant Number
In mathematics, a superabundant number is a certain kind of natural number. A natural number is called ''superabundant'' precisely when, for all : :\frac < \frac where denotes the sum-of-divisors function (i.e., the sum of all positive s of , including itself). The first few superabundant numbers are . For example, the number 5 is not a superabundant number because for , and 5, the sigma is , and . Superabundant numbers were defined by . Unknown to Alaoglu and Erdős, about 30 pages of Ramanujan's 1915 paper "Highly Com ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
