2 problems

For a prime $p$, let $H(p)$ be the height of its Pratt tree: the tree with root $p$, whose children are the prime factors of $p-1$, constructed recursively, stopping at $p=2$. Let…

For a prime $p$, let $H(p)$ be the height of its Pratt tree, let $\log_2 p=\log\log p$ and $\log_3 p=\log\log\log p$, and write $\pi(x)$ for the number of primes at most $x$. Refin…