Nontotient

Nontotient

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

v · Euler's totient function ϕ(n) · Jordan's totient function Jk(n) · Nontotient · Noncototient · Highly totient number · Sparsely totient number · Highly cototient number

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