On The Tree Structure of Natural Numbers, II

Abstract

Each natural number can be associated with some tree graph. Namely, a natural number $n$ can be factorized as $$ n = p_1^{\alpha_1}\ldots p_k^{\alpha_k},$$ where $p_i$ are distinct prime numbers. Since $\alpha_i$ are naturals, they can be factorized 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 the asymptotic formulas for the sums $$\mathcal{M}(x) = \sum_{p\leqslant x} H(p-1),\ \ \mathcal{H}(x) = \sum_{n\leqslant x}2\uparrow\uparrow H(n),$$ and $$\mathcal{L}(x) = \sum_{n\leqslant x}\dfrac{2\uparrow\uparrow H(n)}{2\uparrow\uparrow H(n+1)},$$ where the summation in the first sum is taken over primes.