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 $$n=p_1^{{\alpha }_1}\dots p_k^{{\alpha }_k},$$ 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 $$\mathcal{M}(x)=\sum _{p⩽x}H(p-1),\mathcal{H}(x)=\sum _{n⩽x}2\uparrow \uparrow H(n),$$ and $$\mathcal{L}(x)=\sum _{n⩽x}\frac{2\uparrow \uparrow H(n)}{2\uparrow \uparrow H(n+1)},$$ where the summation in the first sum is taken over primes.

23 pages, 2 figures