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Noncototient
In mathematics, a noncototient is a positive integer ''n'' that cannot be expressed as the difference between a positive integer ''m'' and the number of coprime integers below it. That is, ''m'' − φ(''m'') = ''n'', where φ stands for Euler's totient function, has no solution for ''m''. The '' cototient'' of ''n'' is defined as ''n'' − φ(''n''), so a noncototient is a number that is never a cototient. It is conjectured that all noncototients are even. This follows from a modified form of the slightly stronger version of the Goldbach conjecture: if the even number ''n'' can be represented as a sum of two distinct primes ''p'' and ''q,'' then :pq - \varphi(pq) = pq - (p-1)(q-1) = p+q-1 = n-1. \, It is expected that every even number larger than 6 is a sum of two distinct primes, so probably no odd number larger than 5 is a noncototient. The remaining odd numbers are covered by the observations 1=2-\phi(2), 3 = 9 - \phi(9) and 5 = 25 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
116 (number)
116 (one hundred ndsixteen) is the natural number following 115 and preceding 117. In mathematics 116 is a noncototient, meaning that there is no solution to the equation , where stands for 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 .... 116! + 1 is a factorial prime. There are 116 ternary Lyndon words of length six, and 116 irreducible polynomials of degree six over a three-element field, which form the basis of a free Lie algebra of dimension 116. There are 116 different ways of partitioning the numbers from 1 through 5 into subsets in such a way that, for every ''k'', the union of the first ''k'' subsets is a consecutive sequence of integers. There are 116 different 6×6 Costas arrays.. See also * 116 (other) References {{DE ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
86 (number)
86 (eighty-six) is the natural number following 85 and preceding 87. In mathematics 86 is: * nontotient and a noncototient. * the 25th distinct semiprime and the 13th of the form (2×q). * an Erdős–Woods number, since it is possible to find sequences of 86 consecutive integers such that each inner member shares a factor with either the first or the last member. * a happy number and a self number in base 10. It appears in the Padovan sequence, preceded by the terms 37, 49, 65 (it is the sum of the first two of these). It is conjectured that 86 is the largest n for which the decimal expansion of 2n contains no 0. 86 = (8 × 6 = 48) + (4 × 8 = 32) + (3 × 2 = 6). That is, 86 is equal to the sum of the numbers formed in calculating its multiplicative persistence. In science * 86 is the atomic number of radon. * There are 86 metals on the modern periodic table. In other fields *In American English, and particularly in the food service industry, 86 has become a slang term re ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
154 (number)
154 (one hundred ndfifty-four) is the natural number following 153 and preceding 155. In mathematics 154 is a nonagonal number. Its factorization makes 154 a sphenic number There is no integer with exactly 154 coprimes below it, making 154 a noncototient, nor is there, in base 10, any integer that added up to its own digits yields 154, making 154 a self number 154 is the sum of the first six factorials, if one starts with 0! and assumes that 0!=1. With just 17 cuts, a pancake can be cut up into 154 pieces ( Lazy caterer's sequence). The distinct prime factors of 154 add up to 20, and so do the ones of 153, hence the two form a Ruth-Aaron pair. 154! + 1 is a factorial prime. In music * 154 is an album by Wire, named for the number of live gigs Wire had performed at that time In the military * was a United States Navy ''Trefoil''-class concrete barge during World War II * was a United States Navy ''Admirable''-class minesweeper during World War II * was a United States ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
146 (number)
146 (one hundred [and] forty-six) is the natural number following 145 (number), 145 and preceding 147 (number), 147. In mathematics 146 is an octahedral number, the number of spheres that can be packed into in a regular octahedron with six spheres along each edge. For an octahedron with seven spheres along each edge, the number of spheres on the surface of the octahedron is again 146. It is also possible to arrange 146 disks in the plane into an irregular octagon with six disks on each side, making 146 an octo number. There is no integer with exactly 146 coprimes less than it, so 146 is a nontotient. It is also never the difference between an integer and the total of coprimes below it, so it is a noncototient. And it is not the sum of proper divisors of any number, making it an untouchable number. There are 146 connected partially ordered sets with four labeled elements. See also * 146 (other) References {{Integers, 1 Integers ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
134 (number)
134 (one hundred ndthirty-four) is the natural number following 133 and preceding 135. In mathematics 134 is a nontotient since there is no integer with exactly 134 coprimes below it. And it is a noncototient since there is no integer with 134 integers with common factors below it. 134 is _8C_1 + _8C_3 + _8C_4. In Roman numerals, 134 is a Friedman number since CXXXIV = XV * (XC/X) - I. In the military * was a ''Mission Buenaventura''-class fleet oiler during World War II * was a United States Navy during World War II * was a United States Navy between World War I and World War II * was the lead ship of the United States Navy heavy cruisers during World War II * was a United States Navy ''General G. O. Squier''-class transport ship during World War II * was a United States Navy converted steel-hulled trawler, during World War II * was a United States Navy which saw battle during the Battle of Midway * was a United States Navy during World War II * , was ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
130 (number)
130 (one hundred ndthirty) is the natural number following 129 and preceding 131. In mathematics 130 is a sphenic number. It is a noncototient since there is no answer to the equation ''x'' - φ(''x'') = 130. 130 is the only integer that is the sum of the squares of its first four divisors, including 1: 12 + 22 + 52 + 102 = 130. 130 is the largest number that cannot be written as the sum of four hexagonal numbers. 130 equals both 27 + 2 and 53 + 5 and is therefore a ''doubly strictly '' number. In religion The Book of Genesis states Adam had Seth at the age of 130. The Second Book of Chronicles says that Jehoiada died at the age of 130. In other fields One hundred ndthirty is also: * The year AD 130 or 130 BC * The 130 nanometer process is a semiconductor process technology by semiconductor companies * A 130-30 fund or a ratio up to 150/50 is a type of collective investment vehicle * The C130 Hercules aircraft References See also * List of highways numbered ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
122 (number)
122 (one hundred ndtwenty-two) is the natural number following 121 and preceding 123. In mathematics 122 is a nontotient since there is no integer with exactly 122 coprimes below it. Nor is there an integer with exactly 122 integers with common factors below it, making 122 a noncototient. 122 is a semiprime. 122 is the sum of squares of the divisors of 11. φ(122) = φ(σ(122)). In telephony * The fire emergency telephone number in Austria * The police emergency telephone number in Egypt * The traffic emergency telephone number in China * The police emergency telephone number in Bosnia and Herzegovina In other fields 122 is also: * The atomic number of the chemical element unbibium * The number of men of Michmas at the census (Bible, Nehemiah 7:31) * The Enroute Flight Advisory Service (EFAS) " Flight Watch" frequency: 122.0 MHz * The oldest age in years to which a human being has ever been authenticated to live (Jeanne Calment). See also * List of highways numbered 12 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
100 (number)
100 or one hundred ( Roman numeral: C) is the natural number following 99 and preceding 101. In medieval contexts, it may be described as the short hundred or five score in order to differentiate the English and Germanic use of "hundred" to describe the long hundred of six score or 120. In mathematics 100 is the square of 10 (in scientific notation it is written as 102). The standard SI prefix for a hundred is "hecto-". 100 is the basis of percentages (''per cent'' meaning "per hundred" in Latin), with 100% being a full amount. 100 is a Harshad number in decimal, and also in base-four, a base in-which it is also a self-descriptive number. 100 is the sum of the first nine prime numbers, from 2 through 23. It is also divisible by the number of primes below it, 25. 100 cannot be expressed as the difference between any integer and the total of coprimes below it, making it a noncototient. 100 has a reduced totient of 20, and an Euler totient of 40. A totient valu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
58 (number)
58 (fifty-eight) is the natural number following 57 and preceding 59. In mathematics Fifty-eight is an 11- gonal number, and a Smith number. Given 58, the Mertens function returns 0. 58 is the smallest integer whose square root has a continued fraction with period 7. 58 is equal to the sum of the first seven consecutive prime numbers: 2 + 3 + 5 + 7 + 11 + 13 + 17 = 58. This is a difference of 1 from the 17th prime number and 7th super-prime, 59. There is no solution to the equation ''x'' – φ(''x'') = 58, making 58 a noncototient. However, the sum of the totient function for the first thirteen integers is 58. The regular icosahedron produces 58 distinct stellations. It's dual polyhedron, the regular dodecahedron, only produces three stellations, while the regular octahedron produces a single one, the stella octangula. Neither the tetrahedron nor the cube produce of any of these star polyhedra, which makes the icosahedron the Platonic solid with the most stellation ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
222 (number)
222 (two hundred ndtwenty-two) is the natural number following 221 and preceding 223. In mathematics It is a decimal repdigit and a strobogrammatic number (meaning that it looks the same turned upside down on a calculator display). It is one of the numbers whose digit sum in decimal is the same as it is in binary. 222 is a noncototient, meaning that it cannot be written in the form ''n'' − φ(''n'') where φ is Euler's totient function counting the number of values that are smaller than ''n'' and relatively prime to it. There are exactly 222 distinct ways of assigning a meet and join operation to a set of ten unlabelled elements in order to give them the structure of a lattice, and exactly 222 different six-edge polystick In recreational mathematics, a polystick (or polyedge) is a polyform with a line segment (a 'stick') as the basic shape. A polystick is a connected set of segments in a regular grid. A square polystick is a connected subset of a regu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
52 (number)
52 (fifty-two) is the natural number following 51 and preceding 53. In mathematics Fifty-two is * the 5th Bell number, the number of ways to partition a set of 5 objects. * a decagonal number. * an untouchable number, since it is never the sum of proper divisors of any number, and it is a noncototient since it is not equal to ''x'' − φ(''x'') for any ''x''. * a vertically symmetrical number. In science *The atomic number of tellurium Astronomy *Messier object M52, a magnitude 8.0 open cluster in the constellation Cassiopeia, also known as NGC 7654. *The New General Cataloguebr>object NGC 52, a spiral galaxy in the constellation Pegasus. In other fields Fifty-two is: *The approximate number of weeks in a year. 52 weeks is 364 days, while the tropical year is 365.24 days long. According to ISO 8601, most years have 52 weeks while some have 53. *A significant number in the Maya calendar *On the modern piano, the number of white keys (notes in the C major scale) *The ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
50 (number)
50 (fifty) is the natural number following 49 and preceding 51. In mathematics Fifty is the smallest number that is the sum of two non-zero square numbers in two distinct ways: 50 = 12 + 72 = 52 + 52. It is also the sum of three squares, 50 = 32 + 42 + 52, and the sum of four squares, 50 = 62 + 32 + 22 + 12. It is a Harshad number. 50 is a Stirling number of the first kind: \leftright= 50 and also a Narayana number: \operatorname(6, 3) = \operatorname(6, 4) = 50 There is no solution to the equation φ(''x'') = 50, making 50 a nontotient. Nor is there a solution to the equation ''x'' − φ(''x'') = 50, making 50 a noncototient. In science *The atomic number of tin *The fifth magic number in nuclear physics In religion *In Kabbalah, there are 50 Gates of Wisdom (or Understanding) and 50 Gates of Impurity *The traditional number of years in a jubilee period.Leviticus 25:10 *The Christian Feast of Pentecost takes place on the 50th day of the Easter Season *The Jewish Pent ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
