论文标题
库存循环（即计数序列）有前期$ 2 \ max s_1+60 $
Inventory Loops (i.e. Counting Sequences) have Pre-period $2\max S_1+60$
论文作者
论文摘要
库存序列$（s_0，s_1，s_2，...）$是地图$ f $的迭代，即通过将整数放入其数字描述（例如$ f（1381）= 211318 $，因为“ $ 1381 $”具有两个$ 1 $ 1 $ 3 $，一个$ 3 $，一个$ 8 $）。我们的工作分析了无限基础下的迭代。众所周知，正数的任何起始值最终都是周期性的[1]（例如$ s_0 = 1381 $ reste 1-cycle $ f（31223331418）= 31223331418 $）。所有可能的周期的参数化也已知[2,3]。我们回答布朗斯坦和弗雷恩克尔（Fraenkel）的26年公开问题，显示任何此类起始价值的前期不超过$ 200万美元+60美元，其中$ m = \ max s_1 $。奇怪的是，只有$ o（\ log \ log m）$迭代才能确定周期的周期。
An Inventory Sequence $(S_0, S_1, S_2, ...)$ is the iteration of the map $f$ defined roughly by taking an integer to its numericized description (e.g. $f(1381)=211318$ since "$1381$" has two $1$'s, one $3$, and one $8$). Our work analyzes the iteration under the infinite base. Any starting value of positive digits is known to be ultimately periodic [1] (e.g. $S_0=1381$ reaches the 1-cycle $f(3122331418)=3122331418$). Parametrizations of all possible cycles are also known [2,3]. We answer Bronstein and Fraenkel's open question of 26 years showing the pre-period of any such starting value is no more than $2M+60$ where $M=\max S_1$. And oddly the period of the cycle can be determined after only $O(\log\log M)$ iterations.
