Noncototient

A noncototient is a positive integer n that can not 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.

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 $pq-\phi (pq)=pq-(p-1)(q-1)=p+q-1=n-1.$ It is expected that every even number larger than 6 is a sum of distinct primes, so probably no odd number larger than 5 is a noncototient. The remaining odd numbers are covered by the observations $1=2-\phi (2),3=9-\phi (9)$ and $5=25-\phi (25)$ .

The first few noncototients are:

10, 26, 34, 50, 52, 58, 86, 100, 116, 122, 130, 134, 146, 154, 170, 172, 186, 202, 206, 218, 222, 232, 244, 260, 266, 268, 274, 290, 292, 298, 310, 326, 340, 344, 346, 362, 366, 372, 386, 394, 404, 412, 436, 466, 470, 474, 482, 490, 518, 520

Erdős and Sierpinski asked whether there exist infinitely many noncototients. This was finally answered in the affirmative by Browkin and Schinzel (1995), who showed every member of the infinite family $2^{k}\cdot 509203$ is an example. Since then other infinite families, of roughly the same form, have been given by Flammenkamp and Luca. However, it remains unknown whether or not the set of noncototients possesses a positive lower density.