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225 (number)
225 (two hundred ndtwenty-five) is the natural number following 224 and preceding 226. In mathematics 225 is the smallest number that is a polygonal number in five different ways. It is a square number , an octagonal number, and a squared triangular number . As the square of a double factorial, counts the number of permutations of six items in which all cycles have even length, or the number of permutations in which all cycles have odd length. And as one of the Stirling numbers of the first kind, it counts the number of permutations of six items with exactly three cycles. 225 is a highly composite odd number, meaning that it has more divisors than any smaller odd numbers. After 1 and 9, 225 is the third smallest number ''n'' for which , where ''σ'' is the sum of divisors function and ''φ'' is Euler's totient function. 225 is a refactorable number. 225 is the smallest square number to have one of every digit in some number base (225 is 3201 in base 4) 225 is the fir ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
224 (number)
224 (two hundred ndtwenty-four) is the natural number following 223 and preceding 225. In mathematics 224 is a practical number, and a sum of two positive cubes . It is also , making it one of the smallest numbers to be the sum of distinct positive cubes in more than one way. 224 is the smallest ''k'' with λ(''k'') = 24, where λ(''k'') is the Carmichael function. The mathematician and philosopher Alex Bellos suggested in 2014 that a candidate for the lowest uninteresting number would be 224 because it was, at the time, "the lowest number not to have its own page on he English-language version ofWikipedia". In other areas In the SHA-2 family of six cryptographic hash functions, the weakest is SHA-224, named because it produces 224-bit hash values. It was defined in this way so that the number of bits of security it provides (half of its output length, 112 bits) would match the key length of two-key Triple DES. The ancient Phoenician shekel was a standardized measure of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Divisor Function
In mathematics, and specifically in number theory, a divisor function is an arithmetic function related to the divisors of an integer. When referred to as ''the'' divisor function, it counts the ''number of divisors of an integer'' (including 1 and the number itself). It appears in a number of remarkable identities, including relationships on the Riemann zeta function and the Eisenstein series of modular forms. Divisor functions were studied by Ramanujan, who gave a number of important Modular arithmetic, congruences and identity (mathematics), identities; these are treated separately in the article Ramanujan's sum. A related function is the divisor summatory function, which, as the name implies, is a sum over the divisor function. Definition The sum of positive divisors function σ''z''(''n''), for a real or complex number ''z'', is defined as the summation, sum of the ''z''th Exponentiation, powers of the positive divisors of ''n''. It can be expressed in Summation#Capital ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
New Model Army (band)
New Model Army are an English Rock music, rock band formed in Bradford, West Yorkshire, in 1980 by lead vocalist, guitarist and principal songwriter Justin Sullivan, bassist Stuart Morrow and drummer Phil Tompkins. Sullivan has been the only continuous member of the band, which has seen numerous line-up changes in its four-decade history. Their music draws on influences across the musical spectrum, from punk rock, punk and folk music, folk to soul music, soul, metal music, metal and classical music, classical. Sullivan's lyrics, which range from directly political through to spiritual and personal, have always been considered as a key part of the band's appeal. Whilst having their roots in punk rock, the band have always been difficult to categorise. In 1999, when asked about this, Sullivan said, "We've been labelled as punks, post-punks, Gothic rock, Goth, metal, folk – the lot, but we've always been beyond those style confines". Following a large turnover of personnel, bot ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
225 Winchester
5 (five) is a number, numeral and digit. It is the natural number, and cardinal number, following 4 and preceding 6, and is a prime number. It has attained significance throughout history in part because typical humans have five digits on each hand. In mathematics 5 is the third smallest prime number, and the second super-prime. It is the first safe prime, the first good prime, the first balanced prime, and the first of three known Wilson primes. Five is the second Fermat prime and the third Mersenne prime exponent, as well as the third Catalan number, and the third Sophie Germain prime. Notably, 5 is equal to the sum of the ''only'' consecutive primes, 2 + 3, and is the only number that is part of more than one pair of twin primes, ( 3, 5) and (5, 7). It is also a sexy prime with the fifth prime number and first prime repunit, 11. Five is the third factorial prime, an alternating factorial, and an Eisenstein prime with no imaginary part and real part of the form 3p ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
225 BC
__NOTOC__ Year 225 BC was a year of the pre-Julian Roman calendar. At the time it was known as the Year of the Consulship of Papus and Regulus (or, less frequently, year 529 ''Ab urbe condita''). The denomination 225 BC for this year has been used since the early medieval period, when the Anno Domini calendar era became the prevalent method in Europe for naming years. Events By place Roman Republic * A coalition of Cisalpine Gallic tribes (Taurini, Taurisces, Insubres, Lingones, Salasses, Agones, and Boii), reinforced by large numbers of Transalpine adventurers called Gaesatae (Gaesati), invade Italy. Avoiding the Romans at Ariminum, the Gauls cross the Apennines into Etruria and plunder the country. * To meet this invasion, the Romans call on the Insubres' enemies, the Adriatic Veneti, the Patavini, and the Cenomani, who rapidly mobilise defensive forces. These armies are placed under the command of consuls Lucius Aemilius Papus and Gaius Atilius Regulus. After the Bat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
AD 225
__NOTOC__ Year 225 ( CCXXV) was a common year starting on Saturday (link will display the full calendar) of the Julian calendar. At the time, it was known as the Year of the Consulship of Fuscus and Domitius (or, less frequently, year 978 ''Ab urbe condita''). The denomination 225 for this year has been used since the early medieval period, when the Anno Domini calendar era became the prevalent method in Europe for naming years. Events By place Roman Empire * Emperor Alexander Severus marries Sallustia Orbiana, and possibly raises her father Seius Sallustius to the rank of Caesar. By topic Art and Science * The first Christian paintings appear in Rome, decorating the Catacombs. Births * January 20 – Gordian III, Roman emperor (d. 244) * December 31 – Lawrence, Christian martyr (d. 258) * Trieu Thi Trinh, Vietnamese female warrior (d. 248) * Zhong Hui, Chinese general and politician (d. 264) Deaths * Gaius Vettius Gratus Sabinianus, Roman cons ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Refactorable Number
A refactorable number or tau number is an integer ''n'' that is divisible by the count of its divisors, or to put it algebraically, ''n'' is such that \tau(n)\mid n. The first few refactorable numbers are listed in as : 1, 2, 8, 9, 12, 18, 24, 36, 40, 56, 60, 72, 80, 84, 88, 96, 104, 108, 128, 132, 136, 152, 156, 180, 184, 204, 225, 228, 232, 240, 248, 252, 276, 288, 296, ... For example, 18 has 6 divisors (1 and 18, 2 and 9, 3 and 6) and is divisible by 6. There are infinitely many refactorable numbers. Properties Cooper and Kennedy proved that refactorable numbers have natural density zero. Zelinsky proved that no three consecutive integers can all be refactorable. Colton proved that no refactorable number is perfect. The equation \gcd(n,x) = \tau(n) has solutions only if n is a refactorable number, where \gcd is the greatest common divisor function. Let T(x) be the number of refactorable numbers which are at most x. The problem of determining an a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Euler's Totient Function
In number theory, Euler's totient function counts the positive integers up to a given integer that are relatively prime to . It is written using the Greek letter phi as \varphi(n) or \phi(n), and may also be called Euler's phi function. In other words, it is the number of integers in the range for which the greatest common divisor is equal to 1. The integers of this form are sometimes referred to as totatives of . For example, the totatives of are the six numbers 1, 2, 4, 5, 7 and 8. They are all relatively prime to 9, but the other three numbers in this range, 3, 6, and 9 are not, since and . Therefore, . As another example, since for the only integer in the range from 1 to is 1 itself, and . Euler's totient function is a multiplicative function, meaning that if two numbers and are relatively prime, then . This function gives the order of the multiplicative group of integers modulo (the group of units of the ring \Z/n\Z). It is also used for defining the RSA e ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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] |
226 (number)
226 (two hundred ndtwenty-six) is the natural number following 225 and preceding 227. In mathematics 226 is a happy number, and a semiprime ( 2× 113), and a member of Aronson's sequence. At most 226 different permutation pattern In combinatorial mathematics and theoretical computer science, a permutation pattern is a sub-permutation of a longer permutation. Any permutation may be written in one-line notation as a sequence of digits representing the result of applying the p ...s can occur within a single 9-element permutation. In other fields * The number of ages Hanako has been alive. References Integers {{Num-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Stirling Numbers Of The First Kind
In mathematics, especially in combinatorics, Stirling numbers of the first kind arise in the study of permutations. In particular, the Stirling numbers of the first kind count permutations according to their number of cycles (counting fixed points as cycles of length one). The Stirling numbers of the first and second kind can be understood as inverses of one another when viewed as triangular matrices. This article is devoted to specifics of Stirling numbers of the first kind. Identities linking the two kinds appear in the article on Stirling numbers in general. Definitions The original definition of Stirling numbers of the first kind was algebraic: they are the coefficients s(n,k) in the expansion of the falling factorial :(x)_n = x(x-1)(x-2)\cdots(x-n+1) into powers of the variable x: :(x)_n = \sum_^n s(n,k) x^k, For example, (x)_3 = x(x-1)(x - 2) = 1x^3 - 3x^2 + 2x, leading to the values s(3, 3) = 1, s(3, 2) = -3, and s(3, 1) = 2. Subsequently, it was discovered that th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Permutation
In mathematics, a permutation of a set is, loosely speaking, an arrangement of its members into a sequence or linear order, or if the set is already ordered, a rearrangement of its elements. The word "permutation" also refers to the act or process of changing the linear order of an ordered set. Permutations differ from combinations, which are selections of some members of a set regardless of order. For example, written as tuples, there are six permutations of the set , namely (1, 2, 3), (1, 3, 2), (2, 1, 3), (2, 3, 1), (3, 1, 2), and (3, 2, 1). These are all the possible orderings of this three-element set. Anagrams of words whose letters are different are also permutations: the letters are already ordered in the original word, and the anagram is a reordering of the letters. The study of permutations of finite sets is an important topic in the fields of combinatorics and group theory. Permutations are used in almost every branch of mathematics, and in many other fields of scie ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
