Nontotient
Encyclopedia
In number theory
, a nontotient is a positive integer n which is not in the range
of Euler's totient function
φ, that is, for which φ(x) = n has no solution. In other words, n is a nontotient if there is no integer x that has exactly n coprime
s below it. All odd numbers are nontotients, except 1, since it has the solutions x = 1 and x = 2. The first fifty even nontotients are
An even nontotient may be one more than a prime number
, but never one less, since all numbers below a prime number are, by definition, coprime to it. To put it algebraically, for p prime: φ(p) = p − 1. Also, a pronic number
n(n − 1) is certainly not a nontotient if n is prime since φ(p2) = p(p − 1).
Furthermore, a nontotient can't be expressed as the product of numbers of the form p - 1 and their powers.
Number theory
Number theory is a branch of pure mathematics devoted primarily to the study of the integers. Number theorists study prime numbers as well...
, a nontotient is a positive integer n which is not in the range
Range (mathematics)
In mathematics, the range of a function refers to either the codomain or the image of the function, depending upon usage. This ambiguity is illustrated by the function f that maps real numbers to real numbers with f = x^2. Some books say that range of this function is its codomain, the set of all...
of Euler's totient function
Euler's totient function
In number theory, the totient \varphi of a positive integer n is defined to be the number of positive integers less than or equal to n that are coprime to n In number theory, the totient \varphi(n) of a positive integer n is defined to be the number of positive integers less than or equal to n that...
φ, that is, for which φ(x) = n has no solution. In other words, n is a nontotient if there is no integer x that has exactly n coprime
Coprime
In number theory, a branch of mathematics, two integers a and b are said to be coprime or relatively prime if the only positive integer that evenly divides both of them is 1. This is the same thing as their greatest common divisor being 1...
s below it. All odd numbers are nontotients, except 1, since it has the solutions x = 1 and x = 2. The first fifty even nontotients are
- 1414 (number)14 is the natural number following 13 and preceding 15.In speech, the numbers 14 and 40 are often confused. When carefully enunciated, they differ in which syllable is stressed: 14 vs 40...
, 2626 (number)26 is the natural number following 25 and preceding 27.- In mathematics :26 is the only positive integer that is one greater than a square and one less than a cube .A rhombicuboctahedron has twenty-six sides....
, 3434 (number)34 is the natural number following 33 and preceding 35.-In mathematics:34 is the ninth distinct semiprime and has four divisors including unity and itself. Its neighbors, 33 and 35, also are distinct semiprimes, having four divisors each, and 34 is the smallest number to be surrounded by numbers...
, 3838 (number)This article discusses the number thirty-eight. For the year 38 CE, see 38. For other uses of 38, see 38 38 is the natural number following 37 and preceding 39.-In mathematics:...
, 5050 (number)This article discusses the number fifty. For the year 50 CE, see 50. For other uses of 50, see 50 50 is the natural number following 49 and preceding 51.-In mathematics:...
, 6262 (number)62 is a natural number following 61 and preceding 63. It is a composite number with the divisors 2 and 31 being the eighteenth discreet semiprime. 62 has an aliquot sum of 34; itself a discrete semiprime, and the 10 member aliquot sequence . 62 is therefore the tenth member of the 7-aliquot tree...
, 6868 (number)68 is the natural number following 67 and preceding 69.-In mathematics:Sixty-eight is a nontotient. It is also a Perrin number, preceded in the sequence by 29, 39, 51...
, 7474 (number)74 is the natural number following 73 and preceding 75.A seventy-four was a third-rate warship with 74 guns.-In mathematics:Seventy-four is the twenty-first distinct semiprime and the eleventh of the form 2×q...
, 7676 (number)76 is the natural number following 75 and preceding 77.-In mathematics:Seventy-six is a Lucas number, an automorphic number, a nontotient, a 14-gonal number, and a centered pentagonal number....
, 8686 (number)86 is the natural number following 85 and preceding 87.-In mathematics:86 is the 25th distinct semiprime and the 13th of the form...
, 9090 (number)90 is the natural number preceded by 89 and followed by 91.- In mathematics :*Because 90 is the sum of its unitary divisors , it is a unitary perfect number, and because it is equal to the sum of a subset of its divisors, it is a semiperfect number. 90 is a pronic number. But it is also a...
, 9494 (number)94 is the natural number following 93 and preceding 95.-In mathematics:Ninety-four is the twenty-ninth distinct semiprime and the fourteenth of the form...
, 9898 (number)98 is the natural number following 97 and preceding 99.-In astronomy:* Messier 98, a magnitude 11.0 spiral galaxy in the constellation Coma Berenices....
, 114114 (number)114 is the natural number following 113 and preceding 115.-In mathematics:*One hundred [and] fourteen is an abundant number, a sphenic number and a Harshad number. It is the sum of the first four hyperfactorials, including H...
, 118118 (number)118 is the natural number following 117 and preceding 119.-In mathematics:There is no answer to the equation φ = 118, making 118 a nontotient.118 is the smallest n such that the range n, n + 1, .....
, 122122 (number)122 is the natural number following 121 and preceding 123.-In mathematics:It is a nontotient since there is no integer with exactly 122 coprimes below it...
, 124124 (number)124 is the natural number following 123 and preceding 125.-In mathematics:124 is the sum of eight consecutive primes...
, 134134 (number)134 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...
, 142142 (number)142 is the natural number following 141 and preceding 143.-In astrophysics:* 142 Polana is a dark Main belt asteroid* HD 142 is a 6th magnitude star in the Phoenix constellation-In business and accounting:...
, 146146 (number)146 is the natural number following 145 and preceding 147.-In mathematics:146 is an octahedral number as well as a composite number....
, 152152 (number)152 is the natural number following 151 and preceding 153.-In mathematics:152 is the sum of four consecutive primes...
, 154154 (number)One hundred and fifty-four is the natural number following one hundred and fifty-three and preceding one hundred and fifty-five.-In mathematics:* 154 is a nonagonal number...
, 158158 (number)158 is an even whole 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....
, 170170 (number)170 is the natural number following 169 and preceding 171.-In mathematics:170 is the smallest n for which φ and σ are both square. But 170 is never a solution for φ, making it a nontotient...
, 174174 (number)174 is the natural number following 173 and preceding 175.-In mathematics:* 174 is an even number* 174 is an abundant number with the abundance of 12* 174 is a composite number* 174 is a nontotient number* 174 is an odious number...
, 182182 (number)182 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 154 is less than 182...
, 186186 (number)186 is the natural number following 185 and preceding 187.-In mathematics:* 186 is an even number* 186 is an abundant number, as 198 is greater than 186* 186 is a composite number* There is no integer with exactly 186 coprimes less than it...
, 188188 (number)188 is the natural number following 187 and preceding 189.-In mathematics:* 188 is an even number* 188 has 6 divisors: 1, 2, 4, 47, 94, 188* 188 is a composite number* 188 is a deficient number, as 148 is less than 188* 188 is a happy number...
, 194194 (number)194 is the natural number following 193 and preceding 195.-In mathematics:* 194 is an even number* 194 is a composite number* 194 is a deficient number, as 100 is less than 194...
, 202, 206, 214, 218, 230, 234, 236, 242242 (number)242 is the natural number following 241 and preceding 243.-In mathematics:242 is the smallest integer to start a run of four consecutive integers with the same number of divisors.242 is a nontotient since there is no integer with 242 coprimes below it....
, 244, 246, 248, 254, 258, 266, 274, 278, 284284 (number)Two hundred eighty-four is the natural number following 283 and preceding 285.Its divisors are 1, 2, 4, 71, and 142, adding up to 220, in turn, the divisors of 220 add up to 284, making the two a pair of amicable numbers....
, 286, 290, 298, 302
An even nontotient may be one more than a prime number
Prime number
A prime number is a natural number greater than 1 that has no positive divisors other than 1 and itself. A natural number greater than 1 that is not a prime number is called a composite number. For example 5 is prime, as only 1 and 5 divide it, whereas 6 is composite, since it has the divisors 2...
, but never one less, since all numbers below a prime number are, by definition, coprime to it. To put it algebraically, for p prime: φ(p) = p − 1. Also, a pronic number
Pronic number
A pronic number, oblong number, rectangular number or heteromecic number, is a number which is the product of two consecutive integers, that is, n . The n-th pronic number is twice the n-th triangular number and n more than the n-th square number...
n(n − 1) is certainly not a nontotient if n is prime since φ(p2) = p(p − 1).
Furthermore, a nontotient can't be expressed as the product of numbers of the form p - 1 and their powers.