Extended dataset and Mathematica notebook produced by the Integer sequence (A363743): a(n) = floor(sqrt(log_10(n!))).

This integer sequence was registered and published in the On-Line Encyclopedia of Integer Sequences (OEIS.org) Database on August 17 - 2023, under the OEIS code: A363743. 
This sequence can be generally expressed as follows: a(n) = floor(sqrt(log_10(n!))), where n is a non-negative integer. It should be noted that the aforementioned formula is written in accordance with OEIS specific style sheet format. On the other hand, it was possible to represent the general formula on another two forms that the following: 
1) a(n) = floor(sqrt(A034886(n) - 1)).
 2) a(n) = A000196(A034886(n) -1). 
This dataset verifies the reported properties in the comments section of the OEIS publication. Our evaluation ranges from 0 to n = 5000, in contrast to the publication which ranges from 0 to n = 92. These mentioned properties are the following: 
* Every non-negative integer occurs at least 4-times. 
* Each integer k > 14 appears fewer than k times. 
* The only integers k that occur exactly k times are 11, 13 and 14. 
* This sequence can produce random values between 0 and 1 if we do a(n)/a(n+m) for any non-negative integer m. 
The numerical data showed on this dataset was generated by the following Mathematica program: Array[Floor@ Sqrt[Log10[#!]] &, 5000, 0] The previous program was builded on Mathematica v13.3.0.
 Note: More mathematical details, graphics and technical information can be found in the notebook or .nb file provided in this dataset.

本整数序列于2023年8月17日登记并发表于在线整数序列百科全书（On-Line Encyclopedia of Integer Sequences, OEIS.org）数据库，对应的OEIS编号为A363743。
该序列的通用表达式如下：$a(n) = lfloorsqrt{log_{10}(n!)}
floor$，其中$n$为非负整数，此处$lfloor cdot
floor$代表向下取整（floor）操作。需说明，上述公式严格遵循OEIS官方格式规范书写。
此外，该通用公式还可通过另外两种形式表达：
1) $a(n) = lfloorsqrt{A034886(n) - 1}
floor$。
2) $a(n) = A000196(A034886(n) - 1)$。
本数据集用于验证该OEIS条目标注评论区中提及的序列性质。本次评测的覆盖范围为$n$从0至5000，相较原发表的$n$取值范围0至92有所扩展。上述序列性质具体如下：
* 所有非负整数均至少出现4次。
* 每个大于14的整数$k$的出现次数均少于$k$次。
* 仅整数11、13与14的出现次数恰好等于其自身数值。
* 若对任意非负整数$m$计算$frac{a(n)}{a(n+m)}$，则该比值可生成0到1之间的随机分布值。
本数据集包含的数值数据通过如下Mathematica程序生成：`Array[Floor@ Sqrt[Log10[#!]] &, 5000, 0]`。上述程序基于Mathematica v13.3.0版本构建。
注：更多数学细节、图表及技术信息可查阅本数据集附带的Mathematica笔记本文件（.nb格式）。