Let's denote $N_m$ as the number of solutions of
$$\varphi(n)=m$$
for all $m\geqslant 2$,
where $\varphi$ is Euler totient function, i.e.
$$N_m:=\#\{n\in \mathbb N,\ \varphi(n)=m\}.$$
We can prove easily that $N_m=0$ if $m$ is odd.
We also know that $N_m<\infty$ for all $m$ (thanks to this).
I plotted the sequence $(N_m)$, and I putted red points when
$$m\equiv 0\pmod{12}.$$
The question.
Why is every "high" point (i.e. corresponding to a high value of $N_m$) a red point?
Let $m\in\Bbb{N}$ be given, and let $n\in\Bbb{N}$ be such that $\varphi(n)=m$. If $m\neq0\pmod{12}$ then either $3\nmid m$ or $4\nmid m$. If $4\nmid m$ then either $n=p^k$ or $n=2p^k$ for some prime $p\equiv3\pmod{4}$, or $n=4$. Similarly, if $3\nmid m$ then $n$ is a product of primes congruent to $2\pmod 3$, or $3$ times such a product. Both conditions are quite restrictive, and so $N_m$ should be small.