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Encyclopedia > Nontotient

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 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, 302, 304, 308, 314, 318

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, φ(p) = p - 1. Also, a heteromecic number n(n - 1) is certainly not a nontotient if n is prime since φ(p2) = p(p - 1).

Results from FactBites:

 Talk:Nontotient - Wikipedia, the free encyclopedia (511 words) It's easily proven that no heteromecic number with its second constituent factor being a prime can be a nontotient. I have a feeling that no heteromecic number is a nontotient, even when neither of the two constituent factors is a prime. But the new term may be used in an actual article only after it has been proven that the language did not have an existing term for the concept, and the new term has proven itself with experts and aficionados alike.
 Nontotient - definition of Nontotient in Encyclopedia (212 words) 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 coprimes below it. All odd numbers are nontotients, except 1, since it has the solutions x = 1 and x = 2.
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