Number that is not in the range of Euler's totient function
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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 ncoprimes 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:
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 numbern(n − 1) is certainly not a nontotient if n is prime since φ(p2) = p(p − 1).
If a natural number n is a totient, n · 2k is a totient for all natural numbers k.
There are infinitely many even nontotient numbers: indeed, there are infinitely many distinct primes p (such as 78557 and 271129, see Sierpinski number) such that all numbers of the form 2ap are nontotient, and every odd number has an even multiple which is a nontotient.