Each natural number can be associated with some tree graph. Namely, a natural number n can be factored as \(\begin{aligned} n = p_1^{\alpha _1}\cdots p_k^{\alpha _k}, \end{aligned}\) where \(p_i\) are distinct prime numbers. Since \(\alpha _i\) are naturals, they can be factored in such a manner as well. This process may be continued, building the “factorization tree” until all the top numbers are 1. Let H(n) be the height of the tree corresponding to the number n, and let the symbol \(\uparrow \uparrow \) denote tetration. In this paper, we derive asymptotic formulas for the sums and where the summation in the first sum is taken over primes.
