Solutions to ϕ(x)=k

Having turned 26367 days old today, I noticed that this number is a member of OEIS A289364:

A289364

Numbers n such that $\varphi (x)=12\times n+2$ is solvable, where $\varphi$ is Euler's totient.

Up to 26367, the members of this sequence are:

0, 9, 42, 180, 285, 414, 567, 945, 1109, 1419, 2310, 2655, 3024, 4275, 4740, 5229, 5742, 8034, 10005, 10710, 12192, 14595, 15444, 16317, 18135, 19080, 20049, 21042, 22306, 26367

Clearly, the numbers $x$ satisfying this equation are not numerous. However, I thought so what? Well, fortunately a link provided in the OEIS entry comments shed some light on the matter. I've taken a screenshot of the relevant part of the paper and this is shown in Figure 1.

Figure 1

So it turns out that $\varphi (x)=k$ does not have many solutions if $n\equiv 2\mathrm{mod}12$ and those solutions appear in OEIS A289364. Moreover, Carmichael's conjecture states that the equation is never uniquely satisfied for any positive integer $k$. This was certainly the case when I checked it out for 26367:$$\begin{array}{rl}\text{if }\varphi (x)& =2\times 26367+2\\ x& =316969\text{ or }633938\end{array}$$The paper delves into why there are so few solutions when $n\equiv 2\mathrm{mod}12$ but I've not gone into that here. Perhaps I can cover that in a later post.