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Erdős–Nicolas Number
In number theory, an Erdős–Nicolas number is a number that is not perfect, but that equals one of the partial sums of its divisors. That is, a number is Erdős–Nicolas number when there exists another number such that : \sum_d=n. The first ten Erdős–Nicolas numbers are : 24, 2016, 8190, 42336, 45864, 392448, 714240, 1571328, 61900800 and 91963648. () They are named after Paul Erdős and Jean-Louis Nicolas, who wrote about them in 1975. See also *Descartes number In number theory, a Descartes number is an odd number which would have been an odd perfect number, if one of its composite factors were prime. They are named after René Descartes who observed that the number would be an odd perfect number if onl ..., another type of almost-perfect numbers References Integer sequences {{numtheory-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Paul Erdős
Paul Erdős ( hu, Erdős Pál ; 26 March 1913 – 20 September 1996) was a Hungarian mathematician. He was one of the most prolific mathematicians and producers of mathematical conjectures of the 20th century. pursued and proposed problems in discrete mathematics, graph theory, number theory, mathematical analysis, approximation theory, set theory, and probability theory. Much of his work centered around discrete mathematics, cracking many previously unsolved problems in the field. He championed and contributed to Ramsey theory, which studies the conditions in which order necessarily appears. Overall, his work leaned towards solving previously open problems, rather than developing or exploring new areas of mathematics. Erdős published around 1,500 mathematical papers during his lifetime, a figure that remains unsurpassed. He firmly believed mathematics to be a social activity, living an itinerant lifestyle with the sole purpose of writing mathematical papers with other mathem ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Jean-Louis Nicolas
Jean-Louis Nicolas is a French number theorist. He is the namesake (with Paul Erdős) of the Erdős–Nicolas numbers, and was a frequent co-author of Erdős, who would take over the desk of Nicolas' wife Anne-Marie (also a mathematician) whenever he would visit. Nicolas is also known for his research on partitions,. and for his unusual proof that there exist infinitely many ''n'' for which :\varphi(n) < e^\frac where is Euler's totient function and γ is : he proved this bound unconditionally by providing two different proofs, one in the case that the Riemann hypothesis holds and another in the ... [...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] |
24 (number)
24 (twenty-four) is the natural number following 23 and preceding 25. The SI prefix for 1024 is yotta (Y), and for 10−24 (i.e., the reciprocal of 1024) yocto (y). These numbers are the largest and smallest number to receive an SI prefix to date. In mathematics 24 is an even composite number, with 2 and 3 as its distinct prime factors. It is the first number of the form 2''q'', where ''q'' is an odd prime. It is the smallest number with exactly eight positive divisors: 1, 2, 3, 4, 6, 8, 12, and 24; thus, it is a highly composite number, having more divisors than any smaller number. Furthermore, it is an abundant number, since the sum of its proper divisors ( 36) is greater than itself, as well as a superabundant number. In number theory and algebra *24 is the smallest 5- hemiperfect number, as it has a half-integer abundancy index: *:1 + 2 + 3 + 4 + 6 + 8 + 12 + 24 = 60 = × 24 *24 is a semiperfect number, since adding up all the proper divisors of 24 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
2016 (number)
2016 is the natural number following 2015 and preceding 2017. In mathematics * 2016 is a triangular number, being 1 + 2 + 3 + ... + 63. Equivalently, \tbinom = 2016. * 2016 is a 24-gonal number and a generalized 28-gonal (icosioctagonal) number . * 2016 has 36 divisors. * 211 − 25 = 2016. * 2016 forms a friendly pair with 360, as \dfrac = \dfrac = \dfrac = 3.25 and \dfrac = \dfrac = \dfrac = 3.25. The number 360 itself is a highly composite number, while 2016, while not highly composite, is highly composite among the positive integers not divisible by five. * 2016 × 2 + 1 = 4033. Although 4033 is not prime, as 4033 = 37 × 109, it is a strong pseudoprime to base 2 . Aside from 2016, the only other numbers below 10,000 with this property are 1023, 1638, 2340, 4160, and 7920. * There are 2016 five-cubes in a nine-cube. * 2016 is an Erdős–Nicolas number because, while not perfect, 2016 is the sum of its first 31 divisors (up to and including 288). * 2016 × 20 = 40,320 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
8190 (number)
__NOTOC__ Year 819 ( DCCCXIX) was a common year starting on Saturday (link will display the full calendar) of the Julian calendar. Events By place Europe * Spring – Emperor Louis I marries Judith of Bavaria in Aachen.Rogers, Barbara, Bernhard W. Scholz, and Nithardus. Carolingian Chronicles, Royal Frankish Annals Nithard's Histories. Ann Arbor: Univ. of Michigan, 1972. Print. She becomes his second wife and Empress of the Franks. Like many of the royal marriages of the time, Judith is selected through a bridal show. * Ljudevit, duke of the Slavs in Lower Pannonia, raises a rebellion against the Frankish Empire. Louis I sends an army led by Cadolah of Friuli, but is defeated by the Pannonian Slavs. * Battle of Kupa: Ljudevit defeats the Frankish forces led by Borna, a vassal of Louis I. He escapes with the help of his elite bodyguard. Ljudevit uses the momentum and invades the Duchy of Croatia. * Nominoe, a noble Briton, is appointed by Louis I as count of Vanne ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Number Theory
Number theory (or arithmetic or higher arithmetic in older usage) is a branch of pure mathematics devoted primarily to the study of the integers and arithmetic function, integer-valued functions. German mathematician Carl Friedrich Gauss (1777–1855) said, "Mathematics is the queen of the sciences—and number theory is the queen of mathematics."German original: "Die Mathematik ist die Königin der Wissenschaften, und die Arithmetik ist die Königin der Mathematik." Number theorists study prime numbers as well as the properties of mathematical objects made out of integers (for example, rational numbers) or defined as generalizations of the integers (for example, algebraic integers). Integers can be considered either in themselves or as solutions to equations (Diophantine geometry). Questions in number theory are often best understood through the study of Complex analysis, analytical objects (for example, the Riemann zeta function) that encode properties of the integers, primes ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Perfect Number
In number theory, a perfect number is a positive integer that is equal to the sum of its positive divisors, excluding the number itself. For instance, 6 has divisors 1, 2 and 3 (excluding itself), and 1 + 2 + 3 = 6, so 6 is a perfect number. The sum of divisors of a number, excluding the number itself, is called its aliquot sum, so a perfect number is one that is equal to its aliquot sum. Equivalently, a perfect number is a number that is half the sum of all of its positive divisors including itself; in symbols, \sigma_1(n)=2n where \sigma_1 is the sum-of-divisors function. For instance, 28 is perfect as 1 + 2 + 4 + 7 + 14 = 28. This definition is ancient, appearing as early as Euclid's ''Elements'' (VII.22) where it is called (''perfect'', ''ideal'', or ''complete number''). Euclid also proved a formation rule (IX.36) whereby q(q+1)/2 is an even perfect number whenever q is a prime of the form 2^p-1 for positive integer p—what is now called a Mersenne prime. Two millennia ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Partial Sum
In mathematics, a series is, roughly speaking, a description of the operation of adding infinitely many quantities, one after the other, to a given starting quantity. The study of series is a major part of calculus and its generalization, mathematical analysis. Series are used in most areas of mathematics, even for studying finite structures (such as in combinatorics) through generating functions. In addition to their ubiquity in mathematics, infinite series are also widely used in other quantitative disciplines such as physics, computer science, statistics and finance. For a long time, the idea that such a potentially infinite summation could produce a finite result was considered paradoxical. This paradox was resolved using the concept of a limit during the 17th century. Zeno's paradox of Achilles and the tortoise illustrates this counterintuitive property of infinite sums: Achilles runs after a tortoise, but when he reaches the position of the tortoise at the beginning of t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
