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Highly Composite Number
__FORCETOC__ A highly composite number is a positive integer with more divisors than any smaller positive integer has. The related concept of largely composite number refers to a positive integer which has at least as many divisors as any smaller positive integer. The name can be somewhat misleading, as the first two highly composite numbers (1 and 2) are not actually composite numbers; however, all further terms are. The late mathematician Jean-Pierre Kahane has suggested that Plato must have known about highly composite numbers as he deliberately chose 5040 as the ideal number of citizens in a city as 5040 has more divisors than any numbers less than it. Ramanujan wrote and titled his paper on the subject in 1915. Examples The initial or smallest 38 highly composite numbers are listed in the table below . The number of divisors is given in the column labeled ''d''(''n''). Asterisks indicate superior highly composite numbers. The divisors of the first 15 highly composite ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Superior Highly Composite Number
In mathematics, a superior highly composite number is a natural number which has the highest ratio of its number of divisors to ''some'' positive power of itself than any other number. It is a stronger restriction than that of a highly composite number, which is defined as having more divisors than any smaller positive integer. The first 10 superior highly composite numbers and their factorization are listed. For a superior highly composite number ''n'' there exists a positive real number ''ε'' such that for all natural numbers ''k'' smaller than ''n'' we have :\frac\geq\frac and for all natural numbers ''k'' larger than ''n'' we have :\frac>\frac where ''d(n)'', the divisor function, denotes the number of divisors of ''n''. The term was coined by Ramanujan (1915). For example, the number with the most divisors per square root of the number itself is 12; this can be demonstrated using some highly composites near 12. \frac\approx 1.414, \frac=1.5, \frac\approx 1.633, \ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Highly Composite Numbers
__FORCETOC__ A highly composite number is a positive integer with more divisors than any smaller positive integer has. The related concept of largely composite number refers to a positive integer which has at least as many divisors as any smaller positive integer. The name can be somewhat misleading, as the first two highly composite numbers (1 and 2) are not actually composite numbers; however, all further terms are. The late mathematician Jean-Pierre Kahane has suggested that Plato must have known about highly composite numbers as he deliberately chose 5040 as the ideal number of citizens in a city as 5040 has more divisors than any numbers less than it. Ramanujan wrote and titled his paper on the subject in 1915. Examples The initial or smallest 38 highly composite numbers are listed in the table below . The number of divisors is given in the column labeled ''d''(''n''). Asterisks indicate superior highly composite numbers. The divisors of the first 15 highly composite ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Highly Composite Number Cuisenaire Rods 6
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 * "Hi ... [...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] |
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 of writing it as a product, or , involve 5 itself. However, 4 is composite because it is a product (2 × 2) in which both numbers are smaller than 4. Primes are central in number theory because of the fundamental theorem of arithmetic: every natural number greater than 1 is either a prime itself or can be factorized as a product of primes that is unique up to their order. The property of being prime is called primality. A simple but slow method of checking the primality of a given number n, called trial division, tests whether n is a multiple of any integer between 2 and \sqrt. Faster algorithms include the Miller–Rabin primality test, which is fast but has a small chance of error, and the AKS primality test, which always pro ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Smooth Number
In number theory, an ''n''-smooth (or ''n''-friable) number is an integer whose prime factors are all less than or equal to ''n''. For example, a 7-smooth number is a number whose every prime factor is at most 7, so 49 = 72 and 15750 = 2 × 32 × 53 × 7 are both 7-smooth, while 11 and 702 = 2 × 33 × 13 are not 7-smooth. The term seems to have been coined by Leonard Adleman. Smooth numbers are especially important in cryptography, which relies on factorization of integers. The 2-smooth numbers are just the powers of 2, while 5-smooth numbers are known as regular numbers. Definition A positive integer is called B-smooth if none of its prime factors are greater than B. For example, 1,620 has prime factorization 22 × 34 × 5; therefore 1,620 is 5-smooth because none of its prime factors are greater than 5. This definition includes numbers that lack some of the smaller prime factors; for example, both 10 and 12 are 5-smooth, even though they miss out the prime factors 3 and 5, resp ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
2520 (number)
2520 (two thousand five hundred twenty) is the natural number following 2519 and preceding 2521. In mathematics 2520 is: *the smallest number divisible by all integers from 1 to 10, i.e., it is their least common multiple. *half of 7! ( 5040), meaning 7 factorial, or 1×2×3×4×5×6×7. *the product of five consecutive numbers, namely 3×4×5×6×7. *a superior highly composite number. *a colossally abundant number. *the last highly composite number which is half of the next highly composite number. *the last highly composite number that is a divisor of all following highly composite numbers. *palindromic in bases 11 (199111), and a repdigit in bases 55, 59 and 62. *a Harshad number in bases 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15 and 16. *the aliquot sum of 1080. *part of the 53-aliquot tree. The complete aliquot sequence starting at 1080 is: 1080, 2520, 6840, 16560, 41472, 82311, 27441, 12209, 451, 53, 1, 0. Factors The factors, also called divisor In mathemat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
840 (number)
840 is the natural number following 839 and preceding 841. Mathematical Properties *It is an even number. *It is a practical number. *It is a congruent number. *It is a highly composite number, with 32 divisors : 1, 2, 3, 4, 5, 6, 7, 8, 10, 12, 14, 15, 20, 21, 24, 28, 30, 35, 40, 42, 56, 60, 70, 84, 105, 120, 140, 168, 210, 280, 420, 840. Since the sum of its divisors (excluding the number itself) 2040 > 840 *it is an abundant number and also a superabundant number, *It is an idoneal number, *It is the least common multiple of 1, 2, 3, 4, 5, 6, 7, 8. *It is the largest number ''k'' such that all coprime quadratic residues modulo ''k'' are squares. In this case, they are 1, 121, 169, 289, 361 and 529. *It is an evil number. *It is a palindrome number and a repdigit In recreational mathematics, a repdigit or sometimes monodigit is a natural number composed of repeated instances of the same digit in a positional number system (often implicitly decimal). The word is a portmanteau of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
720 (number)
720 (seven hundred [and] twenty) is the natural number following 700 (number)#710s, 719 and preceding 700 (number)#720s, 721. It is 6! (6 (number), 6 factorial), a composite number with thirty divisors, more than any number below, making it a highly composite number. It is a Harshad number in every base from binary to decimal. 720 is expressible as the product of consecutive integers in two different ways: , and . There are 49 solutions to the equation Euler's totient function, φ(''x'') = 720, more than any integer below it, making 720 a highly totient number. 720 is a 241-polygonal number, gonal number. In other fields 720 is: * A common vertical display resolution for High-definition television, HDTV (see 720p). * 720° is two full rotations; the term "720" refers to a skateboarding trick. * 720° is also the name of a skateboarding video game. * 720 is a dual area codes 303 and 720, area code in the Denver Metro Area along with 303. * 720° is the sum of all the defect ( ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
360 (number)
360 (three hundred sixty) is the natural number following 359 and preceding 361. In mathematics *The divisors of 360 are 1, 2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 18, 20, 24, 30, 36, 40, 45, 60, 72, 90, 120, 180 and 360, making a total of 24 divisors. *360 is a highly composite number. Not only is 360 highly composite, but it is also one of only 7 numbers such that no number less than twice as much has more divisors; the others are 1, 2, 6, 12, 60, and 2520. . 360 is also a superior highly composite number, a colossally abundant number, a refactorable number and a 5-smooth number. *360 is the smallest number divisible by every natural number from 1 to 10 except 7. *One of 360's divisors is 72, which is the number of primes below it. *The sum of Euler's totient function φ(x) over the first thirty-four integers is 360. *A circle is divided into 360 degrees for the purpose of angular measurement. 360° = 2 π rad is also called a round angle. This choice of unit ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
240 (number)
240 (two hundred ndforty) is the natural number following 239 and preceding 241. In mathematics 240 is: *a semiperfect number. *a concatenation of two of its proper divisors. *a highly composite number since it has 20 divisors total (1, 2, 3, 4, 5, 6, 8, 10, 12, 15, 16, 20, 24, 30, 40, 48, 60, 80, 120, and 240), more than any previous number. *a refactorable number or tau number, since it has 20 divisors and 20 divides 240. *a highly totient number, since it has 31 totient answers, more than any previous integer. *a pronic number since it can be expressed as the product of two consecutive integers, 15 and 16. *palindromic in bases 19 (CC19), 23 (AA23), 29 (8829), 39 (6639), 47 (5547) and 59 (4459). *a Harshad number in bases 2, 3, 4, 5, 6, 7, 9, 10, 11, 13, 14, 15 (and 73 other bases). *the aliquot sum of 120 and 57121. *part of the 12161-aliquot tree. The aliquot sequence starting at 120 is: 120, 240, 504, 1056, 1968, 3240, 7650, 14112, 32571, 27333, 12161, 1, 0. 240 is the sm ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
180 (number)
180 (one hundred ndeighty) is the natural number following 179 and preceding 181. In mathematics 180 is an abundant number, with its proper divisors summing up to 366. 180 is also a highly composite number, a positive integer with more divisors than any smaller positive integer. One of the consequences of 180 having so many divisors is that it is a practical number, meaning that any positive number smaller than 180 that is not a divisor of 180 can be expressed as the sum of some of 180's divisors. 180 is a Harshad number and a refactorable number. 180 is the sum of two square numbers: 122 + 62. It can be expressed as either the sum of six consecutive prime numbers: 19 + 23 + 29 + 31 + 37 + 41, or the sum of eight consecutive prime numbers: 11 + 13 + 17 + 19 + 23 + 29 + 31 + 37. 180 is an Ulam number, which can be expressed as a sum of earlier terms in the Ulam sequence only as 177 + 3. 180 is a 61- gonal number. Half a circle has 180 degrees, and thus a U-turn ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
