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Look up nontotient in Wiktionary, the free dictionary. |
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 coprimes 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
- 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 (sequence A005277 in OEIS)
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.
References
- L. Havelock, A Few Observations on Totient and Cototient Valence from PlanetMath
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