论文标题
在自然数的树结构上，ii
On The Tree Structure of Natural Numbers, II
论文作者
论文摘要
每个自然数字可以与某些树图关联。也就是说，可以将天然数量$ n $分解为$$ n = p_1^{α_1} \ ldots p_k^{α_k}，其中$ p_i $是不同的质量数字。由于$α_i$是自然的，因此它们也可以通过这种方式进行分解。可以继续此过程，建立“分解树”，直到所有最高数字为$ 1 $。令$ h（n）$为与数字$ n $相对应的树的高度，然后让符号$ \ uparrow \ uparrow $表示四分。在本文中，我们得出了总和$$ \ mathcal {m}（x）= \ sum_ {p \ leqslant x} h（p-1），\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ sum_ $ \ MATHCAL {l}（x）= \ sum_ {n \ leqslant x} \ dfrac {2 \ uparrow \ uparrow \ uparrow h（n）} {2 \ uparrow \ uparrow \ uparrow \ uparrow \ uparrow h（n+1）}，$ $在第一个总和中，$ quarties the the the the the第一个总和是由prime征收的。
Each natural number can be associated with some tree graph. Namely, a natural number $n$ can be factorized as $$ n = p_1^{α_1}\ldots p_k^{α_k},$$ where $p_i$ are distinct prime numbers. Since $α_i$ 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.
