On The Tree Structure of Natural Numbers, II
Vitalii V. Iudelevich

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 can be factorized as where are distinct prime numbers. Since 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 . Let be the height of the tree corresponding to the number , and let the symbol denote tetration. In this paper, we derive the asymptotic formulas for the sums and where the summation in the first sum is taken over primes.
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Taxonomy
TopicsGraph Labeling and Dimension Problems · Graph theory and applications · Quantum Computing Algorithms and Architecture
