On The Tree Structure of Natural Numbers, II

TL;DR

This paper explores the tree structure of natural numbers through prime factorization, deriving asymptotic formulas for sums involving the height of these trees and their exponential tetration-based functions.

Contribution

It introduces a novel analysis of the factorization tree height and provides asymptotic formulas for related sums involving prime numbers and natural numbers.

Findings

Derived asymptotic formulas for sums over primes and natural numbers.

Analyzed the growth of the factorization tree height $H(n)$.

Established relationships involving tetration in the context of number factorization.

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.