Noncototient
In mathematics, a noncototient is a positive integer n that cannot be expressed as the difference between a positive integer m and the number of coprime integers below it. That is, m − φ(m) = n, where φ stands for Euler's totient function, has no solution for m. The cototient of n is defined as n − φ(n), so a noncototient is a number that is never a cototient.
It is conjectured that all noncototients are even. This follows from a modified form of the Goldbach conjecture: if the even number n can be represented as a sum of two distinct primes p and q, then
It is expected that every even number larger than 6 is a sum of two distinct primes, so probably no odd number larger than 5 is a noncototient. The remaining odd numbers are covered by the observations 
and 
.
For even numbers, it can be shown
Thus, all even numbers n such that n+2 can be written as (p+1)*(q+1) with p, q primes are cototients.
The first few noncototients are
The cototient of n are
Least k such that the cototient of k is n are
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