Nontotient

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 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

Least k such that the totient of k is n are (start with n = 0, 0 if no such k exists)

Greatest k such that the totient of k is n are (start with n = 0, 0 if no such k exists)

Number of ks such that φ(k) = n are (start with n = 0)

According to Carmichael's conjecture there are no 1's in this sequence except the zeroth term.

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).

If a natural number n is a totient, it can be shown that n*2^k is a totient for all natural number k.

There are infinitely many 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 2^ap are nontotient, and every odd number has a multiple which is a nontotient.

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