Define $z_k$ to be the smallest number $z$, such that
$$z\equiv \phi(\phi(p))\pmod p$$
for every prime $p\le k$.
We can assume, that $k$ itself is prime.
The first few numbers are $z_2=z_3=1$ , $z_5=7$ , $z_7=z_{11}=37$ , $z_{13}=11\ 587$ , $z_{17}=131\ 707$
$z_k$ is prime for $k=5,7,11,13,17,163,197,199,557,587,673,1753,2719,3449$ and no other prime $k\le 10007$
What are the next primes $z_k$ ?
Is $z_k$ prime for infinite many prime numbers $k$ ?