Solutions to \( \phi(x)=k\)

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

A289364

Numbers n such that \( \phi(x) = 12 \times n+2\) is solvable, where \( \phi\) 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 \( \phi(x)=k \) does not have many solutions if \(n \equiv  2 \! \! \! \mod \! \! 12 \) 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:$$\begin{align}\text{if } \phi(x)&=2 \times 26367+2\\x&=316969
\text{ or }633938 \end{align}$$The paper delves into why there are so few solutions when \(n \equiv  2 \! \! \! \mod \! \! 12 \) but I've not gone into that here. Perhaps I can cover that in a later post.