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Nontotient
In number theory, a nontotient is a positive integer ''n'' which is not a totient number: it is not in the range of Euler's totient function φ, that is, the equation φ(''x'') = ''n'' has no solution ''x''. In other words, ''n'' is a nontotient if there is no integer ''x'' that has exactly ''n'' coprimes below it. All odd numbers are nontotients, except 1, since it has the solutions ''x'' = 1 and ''x'' = 2. The first few even nontotients are : 14, 26, 34, 38, 50, 62, 68, 74, 76, 86, 90, 94, 98, 114, 118, 122, 124, 134, 142, 146, 152, 154, 158, 170, 174, 182, 186, 188, 194, 202, 206, 214, 218, 230, 234, 236, 242, 244, 246, 248, 254, 258, 266, 274, 278, 284, 286, 290, 298, ... Least ''k'' such that the totient of ''k'' is ''n'' are (0 if no such ''k'' exists) :1, 3, 0, 5, 0, 7, 0, 15, 0, 11, 0, 13, 0, 0, 0, 17, 0, 19, 0, 25, 0, 23, 0, 35, 0, 0, 0, 29, 0, 31, 0, 51, 0, 0, 0, 37, 0, 0, 0, 41, 0, 43, 0, 69, 0, 47, 0, 65, 0, 0, 0, 53, 0, 81, 0, 87, ... [...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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Totient Number
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 (group theory), order of the multiplicative group of integers modulo n, multiplicative group of integers modulo (the Multiplicati ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
182 (number)
182 (one hundred ndeighty-two) is the natural number following 181 and preceding 183. In mathematics * 182 is an even number * 182 is a composite number, as it is a positive integer with a positive divisor other than one or itself * 182 is a deficient number, as the sum of its proper divisors, 154, is less than 182 * 182 is a member of the Mian–Chowla sequence: 1, 2, 4, 8, 13, 21, 31, 45, 66, 81, 97, 123, 148, 182 * 182 is a nontotient number, as there is no integer with exactly 182 coprimes below it * 182 is an odious number * 182 is a pronic number, oblong number or heteromecic number, a number which is the product of two consecutive integers ( 13 × 14) * 182 is a repdigit in the D'ni numeral system ( 77), and in base 9 ( 222) * 182 is a sphenic number, the product of three prime factors * 182 is a square-free number * 182 is an Ulam number * Divisors of 182: 1, 2, 7, 13, 14, 26, 91, 182 In astronomy * 182 Elsa is a S-type main belt asteroid * ... [...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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
170 (number)
170 (one hundred ndseventy) is the natural number following 169 and preceding 171. In mathematics 170 is the smallest ''n'' for which φ(''n'') and σ(''n'') are both square (64 and 324 respectively). But 170 is never a solution for φ(''x''), making it a nontotient. Nor is it ever a solution to ''x'' - φ(''x''), making it a noncototient. 170 is a repdigit in base 4 (2222) and base 16 (AA), as well as in bases 33, 84, and 169. It is also a sphenic number. 170 is the largest integer for which its factorial can be stored in IEEE 754 double-precision floating-point format. This is probably why it is also the largest factorial that Google's built-in calculator will calculate, returning the answer as 170! = 7.25741562 × 10306. There are 170 different cyclic Gilbreath permutations on 12 elements, and therefore there are 170 different real periodic points of order 12 on the Mandelbrot set The Mandelbrot set () is the set of complex numbers c for which the function ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
158 (number)
158 (one hundred ndfifty-eight) is the natural number following 157 and preceding 159. In mathematics 158 is a nontotient, since there is no integer with 158 coprimes below it. 158 is a Perrin number, appearing after 68, 90, 119. 158 is the number of digits in the decimal expansion of 100 !, the product of all the natural numbers up to and including 100. In the military * was a United States Navy during World War II * was a United States Navy during World War II * was a United States Navy during World War II * was a United States Navy following World War II * was a United States Navy during World War II * was a United States Navy ''Trefoil''-class concrete barge during World War II * was a United States Navy during World War II * was a United States Navy converted yacht patrol vessel during World War I In music * The song 158 by the Indie-rock band Blackbud * The song "Here We Go" (1998) from The Bouncing Souls’ '' Tie One On'' CD includes the lyrics "Me, Sh ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
152 (number)
152 (one hundred [and] fifty-two) is the natural number following 151 (number), 151 and preceding 153 (number), 153. In mathematics 152 is the sum of four consecutive primes (31 + 37 + 41 + 43). It is a nontotient since there is no integer with 152 coprimes below it. 152 is a refactorable number since it is divisible by the total number of divisors it has, and in base 10 it is divisible by the sum of its digits, making it a Harshad number. Recently, the smallest repunit probable prime in base 152 was found, it has 589570 digits. The number of surface points on a 6*6*6 cube is 152. In the military * Focke-Wulf Ta 152 was a Luftwaffe high-altitude Interceptor aircraft, interceptor fighter aircraft during World War II * was a United States Navy during World War II * was a United States Navy during World War II * was a United States Navy auxiliary ship, supply ship during World War II * was a United States Navy during World War II * was a United States Navy ship during Wor ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
146 (number)
146 (one hundred ndforty-six) is the natural number following 145 and preceding 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 set In mathematics, especially order theory, a partially ordered set (also poset) formalizes and generalizes the intuitive concept of an ordering, sequencing, or arran ... [...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 a United ... [...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 Jeanne Louise Calment (; 21 February 1 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
118 (number)
118 (one hundred ndeighteen) is the natural number following 117 and preceding 119. In mathematics There is no answer to the equation φ(''x'') = 118, making 118 a nontotient. Four expressions for 118 as the sum of three positive integers have the same product: :14 + 50 + 54 = 15 + 40 + 63 = 18 + 30 + 70 = 21 + 25 + 72 = 118 and :14 × 50 × 54 = 15 × 40 × 63 = 18 × 30 × 70 = 21 × 25 × 72 = 37800. 118 is the smallest number that can be expressed as four sums with the same product in this way. Because of its expression as , it is a Leyland number of the second kind. 118!! - 1 is a prime number, where !! denotes the double factorial (the product of even integers up to 118). In other fields * There are 118 known elements, the 118th element being oganesson. See also * 118 (other) 118 may refer to: *118 (number) *AD 118 *118 BC *118 (TV series) *118 (film) *118 (Tees) Corps Engineer Regiment *118 (Tees) Field Squadron, Royal Engineers See also *11/8 (disambig ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
114 (number)
114 (one hundred ndfourteen) is the natural number following 113 and preceding 115. In mathematics *114 is an abundant number, a sphenic number and a Harshad number. It is the sum of the first four hyperfactorials, including H(0). At 114, the Mertens function sets a new low of -6, a record that stands until 197. *114 is the smallest positive integer* which has yet to be represented as a3 + b3 + c3, where a, b, and c are integers. It is conjectured that 114 can be represented this way. (*Excluding integers of the form 9k ± 4, for which solutions are known not to exist.) *There is no answer to the equation φ(x) = 114, making 114 a nontotient. *114 appears in the Padovan sequence, preceded by the terms 49, 65, 86 (it is the sum of the first two of these). *114 is 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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
