Let $p$ be an odd prime, and let $f=\Phi_p$ be the $p$-th cyclotomic polynomial. Denote by $S_p$ the set of primes $q$ such that there exists a sequence of primes $p_1,\dots, p_g$ such that $p_1=f(1)=p$, $p_g=q$ and $p_{j+1}|f(p_j)$ for all $j=1,\dots,g-1$. If $g$ is the least possible positive integer for this to hold, we say that $q$ has $p$-generation $g$, and we write $g_p(q)=g$.
It's easy to see that all primes in $S_p$ other than $p$ are congruent to 1 mod p. Now, my intuition was telling me that $S_p\setminus \{p\}$ would in fact be the set of all primes $q \equiv 1 \mod p$, and that this prime divisors tree way of constructing the primes was just a way of presenting such primes in a naturally arising order (generationally speaking). However, when I went on and try to prove this, I realize that I could not even prove that $S_p$ was infinite. So I tried and looked up the literature, but could not find anything that I could use to answer this question.
Hence, first of all I wanted to ask if anybody knows any reference related to this sort of problem and or techniques that might be worth trying to get some more insight in the set $S_p$.
In the case the answer of whether $S_p\setminus \{p\}$ is the set of all primes $q\equiv 1 \mod p$ is actually not known, then in my opinion a good starting point would be to first study the behaviour of the sums $$ \mathfrak{s}_p(g):= \underset{g_p(q)\le g}{\sum_{q\in S_p}} \frac{1}{q}$$ as $g\to \infty$.
Clearly, if $S_p$ turns out to be finite, then $\lim_{g\to \infty}\mathfrak{s}_p(g)$ is finite, not very interesting.
On the other hand, if $S_p\setminus \{p\}$ is the set the set of all primes $q\equiv 1 \mod p$, then we know that $\mathfrak{s}_p(g)$ diverges as $g\to \infty$ from Dirichlet's theorem, but it is not clear what the rate of divergence would be, since the sum grows much slowler compared to the sum of the reciprocals of the primes in the usual order. Denoting by $N(g)$ the largest $N\ge 1$ such that for all primes $q\equiv 1 \mod p, q\le N$ one has $g_p(q)\le g$, then we would get a lower bound that is asymptotic to $\frac{1}{p-1}\log\log(N(g))$. However, first of all the problem of determining the growth of $N(g)$ as $g\to \infty$ seems highly non-trivial. Also, the chance of such a bound being asymptotically tight seems very slim, because it ignores the contribution of (too?) many larger primes along the way.
Finally, if $S_p$ happens to be infinite, but $S_p\setminus \{p\}$ is not the set of all primes $q\equiv 1 \mod p$, then the question of whether $\lim_{g\to \infty}\mathfrak{s}_p(g)$ is finite or infinite becomes non-trivial, and would certainly require some interesting non-trivial insight on the structure of the primes in $S_p$.
Therefore, my second question is if anybody has any ideas or tricks to share that can be useful to provide some non-trivial lower bounds for $\mathfrak{s}_p(g)$. I have tried some elementary considerations, but I haven't got any interesting bound that way. Maybe some technique from analytic number theory that could be applied in this case?
