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Primitive Abundant Number
In mathematics a primitive abundant number is an abundant number whose proper divisors are all deficient numbers. For example, 20 is a primitive abundant number because: :#The sum of its proper divisors is 1 + 2 + 4 + 5 + 10 = 22, so 20 is an abundant number. :#The sums of the proper divisors of 1, 2, 4, 5 and 10 are 0, 1, 3, 1 and 8 respectively, so each of these numbers is a deficient number. The first few primitive abundant numbers are: : 20, 70, 88, 104, 272, 304, 368, 464, 550, 572 ... The smallest odd primitive abundant number is 945. A variant definition is abundant numbers having no abundant proper divisor . It starts: : 12, 18, 20, 30, 42, 56, 66, 70, 78, 88, 102, 104, 114 Properties Every multiple of a primitive abundant number is an abundant number. Every abundant number is a multiple of a primitive abundant number or a multiple of a perfect number. Every primitive abundant number is either a primitive semiperfect number or a weird number In number ... [...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 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 abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of ... [...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] |
Proper Divisor
In mathematics, a divisor of an integer n, also called a factor of n, is an integer m that may be multiplied by some integer to produce n. In this case, one also says that n is a multiple of m. An integer n is divisible or evenly divisible by another integer m if m is a divisor of n; this implies dividing n by m leaves no remainder. Definition An integer is divisible by a nonzero integer if there exists an integer such that n=km. This is written as :m\mid n. Other ways of saying the same thing are that divides , is a divisor of , is a factor of , and is a multiple of . If does not divide , then the notation is m\not\mid n. Usually, is required to be nonzero, but is allowed to be zero. With this convention, m \mid 0 for every nonzero integer . Some definitions omit the requirement that m be nonzero. General Divisors can be negative as well as positive, although sometimes the term is restricted to positive divisors. For example, there are six divisors of 4; they are ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Deficient Number
In number theory, a deficient number or defective number is a number ''n'' for which the sum of divisors of ''n'' is less than 2''n''. Equivalently, it is a number for which the sum of proper divisors (or aliquot sum) is less than ''n''. For example, the proper divisors of 8 are 1, 2, and 4, and their sum is less than 8, so 8 is deficient. Denoting by ''σ''(''n'') the sum of divisors, the value 2''n'' − ''σ''(''n'') is called the number's deficiency. In terms of the aliquot sum ''s''(''n''), the deficiency is ''n'' − ''s''(''n''). 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 ... [...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] |
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] |
70 (number)
70 (seventy) is the natural number following 69 and preceding 71. In mathematics 70 is: * a sphenic number because it factors as 3 distinct primes. * a Pell number. * the seventh pentagonal number. * the fourth tridecagonal number. * the fifth pentatope number. * the number of ways to choose 4 objects out of 8 if order does not matter. This makes it a central binomial coefficient. * the smallest weird number, a natural number that is abundant but not semiperfect. * a palindromic number in bases 9 (779), 13 (5513) and 34 (2234). * a Harshad number in bases 6, 8, 9, 10, 11, 13, 14, 15 and 16. * an Erdős–Woods number, since it is possible to find sequences of 70 consecutive integers such that each inner member shares a factor with either the first or the last member. The sum of the first 24 squares starting from 1 is 70 = 4900, i.e. a square pyramidal number. This is the only non trivial solution to the cannonball problem and relates 70 to the Leech lattice and thus string the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
88 (number)
88 (eighty-eight) is the natural number following 87 and preceding 89. In mathematics 88 is: * a refactorable number. * a primitive semiperfect number. * an untouchable number. * a hexadecagonal number. * an Erdős–Woods number, since it is possible to find sequences of 88 consecutive integers such that each inner member shares a factor with either the first or the last member. * a palindromic number in bases 5 (3235), 10 (8810), 21 (4421), and 43 (2243). * a repdigit in bases 10, 21 and 43. * a 2-automorphic number. * the smallest positive integer with a Zeckendorf representation requiring 5 Fibonacci numbers. * a strobogrammatic number. * the largest number in English not containing the letter 'n' in its name, when using short scale. 88 and 945 are the smallest coprime abundant numbers. In science and technology *The atomic number of the element radium. * The number of constellations in the sky as defined by the International Astronomical Union. * Messier object M88, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
104 (number)
104 (one hundred ndfour) is the natural number following 103 and preceding 105. In mathematics 104 is a primitive semiperfect number and a composite number, with its divisors being 1, 2, 4, 8, 13, 26, 52 and 104. As it has 8 divisors total, and 8 is one of those divisors, 104 is a refactorable number. The distinct prime factors of 104 add up to 15, and so do the ones of 105, hence the two numbers form a Ruth-Aaron pair under the first definition. In regular geometry, 104 is the smallest number of unit line segments that can exist in a plane with four of them touching at every vertex.A figure made up of a row of 4 adjacent congruent rectangles is divided into 104 regions upon drawing diagonals of all possible rectangles φ(104) = φ(σ(104)). In science *The atomic number of rutherfordium. *Number of degrees Fahrenheit corresponding to 40 Celsius. In other fields 104 is also: *The number of Corinthian columns in the Temple of Olympian Zeus, the largest temple ever built ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
12 (number)
12 (twelve) is the natural number following 11 (number), 11 and preceding 13 (number), 13. Twelve is a superior highly composite number, divisible by 2 (number), 2, 3 (number), 3, 4 (number), 4, and 6 (number), 6. It is the number of years required for an Jupiter#Pre-telescopic research, orbital period of Jupiter. It is central to many systems of timekeeping, including the Gregorian calendar, Western calendar and time, units of time of day and frequently appears in the world's major religions. Name Twelve is the largest number with a monosyllable, single-syllable name in English language, English. Early Germanic languages, Germanic numbers have been theorized to have been non-decimal: evidence includes the unusual phrasing of 11 (number), eleven and twelve, the long hundred, former use of "hundred" to refer to groups of 120 (number), 120, and the presence of glosses such as "tentywise" or "ten-count" in medieval texts showing that writers could not presume their readers would no ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
18 (number)
18 (eighteen) is the natural number following 17 and preceding 19. In mathematics * Eighteen is a composite number, its divisors being 1, 2, 3, 6 and 9. Three of these divisors (3, 6 and 9) add up to 18, hence 18 is a semiperfect number. Eighteen is the first inverted square-prime of the form ''p''·''q''2. * In base ten, it is a Harshad number. * It is an abundant number, as the sum of its proper divisors is greater than itself (1+2+3+6+9 = 21). It is known to be a solitary number, despite not being coprime to this sum. * It is the number of one-sided pentominoes. * It is the only number where the sum of its written digits in base 10 (1+8 = 9) is equal to half of itself (18/2 = 9). * It is a Fine number. In science Chemistry * Eighteen is the atomic number of argon. * Group 18 of the periodic table is called the noble gases. * The 18-electron rule is a rule of thumb in transition metal chemistry for characterising and predicting the stability of metal complexes. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
30 (number)
30 (thirty) is the natural number following 29 and preceding 31. In mathematics 30 is an even, composite, pronic number. With 2, 3, and 5 as its prime factors, it is a regular number and the first sphenic number, the smallest of the form , where is a prime greater than 3. It has an aliquot sum of 42, which is the second sphenic number. It is also: * A semiperfect number, since adding some subsets of its divisors (e.g., 5, 10 and 15) equals 30. * A primorial. * A Harshad number in decimal. * Divisible by the number of prime numbers ( 10) below it. * The largest number such that all coprimes smaller than itself, except for 1, are prime. * The sum of the first four squares, making it a square pyramidal number. * The number of vertices in the Tutte–Coxeter graph. * The measure of the central angle and exterior angle of a dodecagon, which is the petrie polygon of the 24-cell. * The number of sides of a triacontagon, which in turn is the petrie polygon of the 120-cell ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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