关于真因数之和的数字问题
On the digits of the sum of proper divisors
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中文总结 AI 辅助
研究整数\(n\)真因数之和\(s(n)\)数字的概率问题，证明其服从本福特定律，表明多数\(n\)的\(s(n)\)前后\(k(x)\)个数字含所有十进制数字，给出合数\(n\)中\(s(n)\)缺数字情况的上界，揭示素数对结果的影响。
中文摘要 AI 辅助
我们研究了几个关于整数\(n\)的真因数之和\(s(n)\)的数字的概率问题。特别地，我们证明了\(s(n)\)关于对数密度服从本福特定律。此外，我们表明，对于每个趋于无穷的函数\(k(x)\)，几乎所有小于等于\(x\)的整数\(n\)，其\(s(n)\)的前\(k(x)\)个数字和后\(k(x)\)个数字中都出现了每一个十进制数字。我们还给出了小于等于\(x\)的合数\(n\)的数量的一个上界，对于这些合数\(n\)，\(s(n)\)在其十进制展开中至少缺少一个数字。这与Benli、Cesana、Dartyge、Dombrowsky和Thompson最近一篇论文的主要结果形成对比，在那篇论文中，输入的\(n\)不要求是合数。事实证明，素数对原像集\(s^{-1}(\mathcal{A})\)有很大贡献，其中\(\mathcal{A}\)是一个缺少数字的整数集。我们对于合数\(n\)的结果表明，排除素数输入时，计数要小得多。
英文摘要
We study several probabilistic questions concerning the digits of $s(n)$, the sum of proper divisors of an integer $n$. In particular, we show that $s(n)$ obeys Benford's law with respect to logarithmic density. Moreover, we show that, for every function $k(x) \rightarrow \infty$, almost all integers $n \leq x$ have every decimal digit occurring among the first $k(x)$ digits and the last $k(x)$ digits of $s(n)$. We also present an upper bound for the number of composite integers $n$ up to $x$ for which $s(n)$ is missing at least one digit in its decimal expansion. This is in contrast with the main result of a recent paper of Benli, Cesana, Dartyge, Dombrowsky, and Thompson, in which the inputs $n$ were not required to be composite. It turns out that the primes make a substantial contribution to the preimage set $s^{-1}(\mathcal{A})$, where $\mathcal{A}$ is a set of integers with missing digits. Our result for composite $n$ shows that the count is much smaller when prime inputs are excluded.
