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Largely Composite Number
A highly composite number is a positive integer that has more divisors than all smaller positive integers. If ''d''(''n'') denotes the number of divisors of a positive integer ''n'', then a positive integer ''N'' is highly composite if ''d''(''N'') > ''d''(''n'') for all ''n'' < ''N''. For example, 6 is highly composite because ''d''(6)=4, and for ''n''=1,2,3,4,5, you get ''d''(''n'')=1,2,2,3,2, respectively, which are all less than 4. A related concept is that of a largely composite number, a positive integer that has at least as many divisors as all smaller positive integers. The name can be somewhat misleading, as the first two highly composite numbers (1 and 2) are not actually ; however, all further terms are. Ramanujan wrote a paper on high ... [...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), ''High'' (The Blue Nile album) or the title song, 2004 * High (Flotsam and Jetsam album), ''High'' (Flotsam and Jetsam album), 1997 * High (New Model Army album), ''High'' (New Model Army album) or the title song, 2007 * High (Royal Headache album), ''High'' (Royal Headache album) or the title song, 2015 * High (Keith Urban album), ''High'' (Keith Urban album), 2024 * High (EP), ''High'' (EP), by Jarryd James, o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
36 (number)
36 (thirty-six) is the natural number following 35 (number), 35 and preceding 37 (number), 37. In mathematics 36 is both the Square number, square of 6, six, and the eighth triangular number or the sum of the first eight non-zero positive integers, which makes 36 the first non-trivial square triangular number. Aside from being the smallest square triangular number other than 1, it is also the only triangular number (other than 1) whose square root is also a triangular number. 36 is also the eighth refactorable number, as it has exactly nine positive divisors, and 9 is one of them; in fact, it is the smallest positive integer with at least nine divisors, which leads 36 to be the 7th highly composite number. It is the sum of the fourth pair of Twin prime, twin-primes (17 (number), 17 + 19 (number), 19), and the 18th Harshad number in Base ten, decimal, as it is divisible by the sum of its digits (9). It is the smallest number n with exactly eight solutions (37 (number), 37, 57 (nu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Prime Factors
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 , called trial division, tests whether is a multiple of any integer between 2 and . 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 produce ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Highly Composite Numbers
A highly composite number is a positive integer that has more divisors than all smaller positive integers. If ''d''(''n'') denotes the number of divisors of a positive integer ''n'', then a positive integer ''N'' is highly composite if ''d''(''N'') > ''d''(''n'') for all ''n'' < ''N''. For example, 6 is highly composite because ''d''(6)=4, and for ''n''=1,2,3,4,5, you get ''d''(''n'')=1,2,2,3,2, respectively, which are all less than 4. A related concept is that of a largely composite number, a positive integer that has at least as many divisors as all smaller positive integers. 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. Ramanujan wrote a paper on highly composite numbers in 1915. Th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Primorial
In mathematics, and more particularly in number theory, primorial, denoted by "", is a function from natural numbers to natural numbers similar to the factorial function, but rather than successively multiplying positive integers, the function only multiplies prime numbers. The name "primorial", coined by Harvey Dubner, draws an analogy to ''primes'' similar to the way the name "factorial" relates to ''factors''. Definition for prime numbers For the th prime number , the primorial is defined as the product of the first primes: :p_n\# = \prod_^n p_k, where is the th prime number. For instance, signifies the product of the first 5 primes: :p_5\# = 2 \times 3 \times 5 \times 7 \times 11= 2310. The first few primorials are: : 1, 2, 6, 30, 210, 2310, 30030, 510510, 9699690... . Asymptotically, primorials grow according to: :p_n\# = e^, where is Little O notation. Definition for natural numbers In general, for a positive integer , its primorial, , is th ... [...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 in which every prime factor is at most 7. Therefore, 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. 2-smooth numbers are simply the Power of two, powers of 2, while 5-smooth numbers are also known as regular numbers. Definition A negative and positive numbers, 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, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
2520 (number)
2520 (two thousand five hundred ndtwenty) is the natural number following 2519 and preceding 2521. In mathematics 2520 is: *the smallest number divisible by all integers from one to ten, i.e., it is their least common multiple. *half of 7! ( 5040), meaning 7 factorial, or 1\times 2\times 3\times 4\times 5\times 6\times 7. *the product of five consecutive numbers, namely 3\times 4\times 5\times 6\times 7. *the 7th superior highly composite number. *the 7th colossally abundant number. *the 18th highly composite number. *the last highly composite number that 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 undecimal (199111) and a repdigit in bases 55, 59, and 62. *a Harshad number in all bases between binary and hexadecimal. *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, 12 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
840 (number)
840 (eight hundred ndforty) 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 the 15th 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 the numbers from 1 to 8. *It is the smallest number divisible by every natural number from 1 to 10, except 9. *It is the number under 1000 with the most divisors, at 32. *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 repdig ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
720 (number)
720 (seven hundred ndtwenty) is the natural number following 719 and preceding 721. In mathematics 720 is: *6! ( 6 factorial). *a composite number with 30 divisors, more than any number below, making it the 14th highly composite number. *a highly abundant number. *a Harshad number in every base from binary to decimal. *the smallest number to be palindromic in 16 bases. *a 241- gonal number. 720 is expressible as the product of consecutive integers in two different ways: and . There are 49 solutions to the equation , more than any integer below it, making 720 a highly totient number. In other fields 720 is: * A common vertical display resolution for 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 code in the Denver Metro Area along with 303. * 720° is the sum of all the defects of any polyhedron In geometry, a polyhedron (: polyhedra or polyhedr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
360 (number)
360 (three hundred [and] sixty) is the natural number following 359 (number), 359 and preceding 361 (number), 361. In mathematics * 360 is the 13th highly composite number and one of only seven numbers such that no number less than twice as much has more divisors; the others are 1, 2, 6, 12 (number), 12, 60 (number), 60, and 2520 (number), 2520 . *360 is also the 6th superior highly composite number, the 6th colossally abundant number, a refactorable number, a 5-smooth number, and a Harshad number in Base ten, decimal since the sum of its digits (9) is a divisor of 360. *360 is divisible by the number of its divisors (24 (number), 24), and it is the smallest number divisible by every natural number from 1 to 10, except 7 (number), 7. Furthermore, one of the divisors of 360 is 72 (number), 72, which is the number of Prime number, primes below it. *360 is the sum of twin primes (179 (number), 179 + 181 (number), 181) and the sum of four consecutive Power of three, powers of thre ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
240 (number)
240 (two hundred ndforty) is the natural number following 239 and preceding 241. Mathematics 240 is a pronic number, since it can be expressed as the product of two consecutive integers, 15 and 16. It is a semiperfect number, equal to the concatenation of two of its proper divisors (24 and 40). It is also the 12th highly composite number, with 20 divisors in total, more than any smaller number; and a refactorable number or tau number, since one of its divisors is 20, which divides 240 evenly. 240 is the aliquot sum of only two numbers: 120 and 57121 (or 2392); and is part of the 12161-aliquot tree that goes: 120, 240, 504, 1056, 1968, 3240, 7650, 14112, 32571, 27333, 12161, 1, 0. It is the smallest number that can be expressed as a sum of consecutive primes in three different ways: \begin 240 & = 113 + 127 \\ 240 & = 53 + 59 + 61 + 67 \\ 240 & = 17 + 19 + 23 + 29 + 31 + 37 + 41 + 43 \\ \end 240 is highly totient, since it has thirty-one totient answers, more than any p ... [...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 the 11th highly composite number, a positive integer with 18 divisors, more 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, while 61 is the 18th prime number. H ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
