On the tree structure of natural numbers. II

European Journal of Mathematics. 2025. Vol. 11. Article 16.

Vitalii Iudelevich

Each natural number can be associated with some tree graph. Namely, 
        a natural number $n$ can be factored as 
        $$ n = p_1^{\alpha_1}\cdots p_k^{\alpha_k},$$
        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 $\uparrow\uparrow$ denotes 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.

Keywords: shifted primes, Tetration, Tree factorization