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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 this sequence: : 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, ... The least value of ''k'' such that the totient of ''k'' is ''n'' are (0 if no such ''k'' exists) are this sequence: :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, ... [...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 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 enc ... [...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 In mathematics, a square-free integer (or squarefree integer) is an integer which is divisible by no square number other than 1. That is, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
86 (number)
86 (eighty-six) is the natural number following 85 (number), 85 and preceding 87 (number), 87. In mathematics 86 is: * nontotient and a noncototient. * the 25th distinct semiprime and the 13th of the form (2.q). * together with 85 (number) , 85 and 87 (number), 87, forms the middle semiprime in the 2nd cluster of three consecutive semiprimes; the first comprising 33 (number), 33, 34 (number), 34, 35 (number), 35. * 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. * with an aliquot sum of 46 (number), 46; itself a semiprime, within an aliquot sequence of seven members (86,46 (number), 46,26 (number), 26,16 (number), 16,15 (number), 15,9 (number), 9,4 (number), 4,3 (number), 3,1 (number), 1,0) in the Prime 3 (number), 3-aliquot tree. It appears in the Padovan sequence, preceded by the terms 37, 49, 65 (it is t ... [...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 a two-dimensional set (mathematics), set that is defined ... [...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 coprime In number theory, two integers and are coprime, relatively prime or mutually prime if the only positive integer that is a divisor of both of them is 1. Consequently, any prime number that divides does not divide , and vice versa. This is equiv ...s below it. 158 is a Perrin number, appearing after 68, 90, 119. References External links The Number 158 {{DEFAULTSORT:158 (Number) Integers ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
152 (number)
152 (one hundred ndfifty-two) is the natural number following 151 and preceding 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 In mathematics, a harshad number (or Niven number) in a given radix, number base is an integer that is divisible by the digit sum, sum of its digits when written in that base. Harshad numbers in base are also known as -harshad (or -Niven) numbers .... The smallest repunit probable prime in base 152 was found in June 2015, it has 589570 digits. The number of surface points on a 6*6*6 cube is 152. References External links Oklahoma Highway 152 {{DEFAULTSORT:152 (Number) Integers ... [...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. 146 is also a repdigit in base 8 (222). See also * 146 (other) References Integers {{Num- ... [...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 numeral Roman numerals are a numeral system that originated in ancient Rome and remained the usual way of writing numbers throughout Europe well into the Late Middle Ages. Numbers are written with combinations of letters from the Latin alphabet, ea ...s, 134 is a Friedman number since CXXXIV = XV * (XC/X) - I. {{DEFAULTSORT:134 (Number) Integers ... [...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 In mathematics, a semiprime is a natural number that is the product of exactly two prime numbers. The two primes in the product may equal each other, so the semiprimes include the squares of prime numbers. Because there are infinitely many prime n .... φ(122) = φ(σ(122)). References Integers {{Num-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
118 (number)
118 (one hundred [and] eighteen) is the natural number following 117 (number), 117 and preceding 119 (number), 119. In mathematics There is no answer to the equation Euler's totient function, 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#Leyland_number_of_the_second_kind, 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). See also * 118 (other) References [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
114 (number)
114 (one hundred [and] fourteen) is the natural number following 113 (number), 113 and preceding 115 (number), 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, Sums of three cubes, 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 Euler's totient function, 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 base 7 (222). See also * 114 (other) References Integers {{Num-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
