Extended dataset 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17-20日在在线整数序列百科全书（On-Line Encyclopedia of Integer Sequences，OEIS.org）数据库中注册并发布，对应的OEIS编号为A363743。该序列的通用表达式为：$a(n) = lfloor sqrt{log_{10}(n!)} floor$，其中$n$为非负整数。需注意，上述公式严格遵循OEIS专属样式表格式规范。此外，该通用公式还可表示为另外两种形式：1) $a(n) = lfloor sqrt{A034886(n) - 1} floor$；2) $a(n) = A000196(A034886(n) - 1)$。本数据集验证了该OEIS发布内容评论区中记载的各项性质。本次评估的取值范围为0至$n=5000$，相较原发布仅覆盖0至$n=92$的范围有所拓展。上述记载的性质如下： * 每个非负整数至少出现4次； * 所有大于14的整数$k$的出现次数均少于$k$次； * 恰好出现$k$次的整数$k$仅为11、13和14； * 若对任意非负整数$m$计算$frac{a(n)}{a(n+m)}$，则该序列可生成介于0与1之间的随机数值。 本数据集附带的数值数据通过以下Mathematica程序生成：`Array[Floor@ Sqrt[Log10[#!]] &, 5000, 0]`。前述程序基于Mathematica v13.3.0版本编写。注：本数据集附带的.nb格式Mathematica笔记本文件中包含更多数学细节、图表及技术信息。