Extended dataset produced by the Integer sequence (A363743): a(n) = floor(sqrt(log_10(n!))).
Description
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.
Files
Steps to reproduce
To calculate any term in this sequence dataset, we utilized Mathematica. Our objective was to evaluate specific term values using any integer value for n. Next, we used Mathematica to compute the sequence for n = 7, n = 431, and n = 3973, and obtained the corresponding values: In[1]:= a[7] = Floor[Sqrt[Log10[7!]]] Out[1]= 1 In[90]:= a[431] = Floor[Sqrt[Log10[431!]]] Out[90]= 30 In[93]:= a[3973] = Floor[Sqrt[Log10[3973!]]] Out[93]= 112 On the other hand, if you wish to evaluate the sequence A363743 for larger ranges than those displayed in this dataset, you simply need to modify the value of n in the following Mathematica code: Array[Floor@ Sqrt[Log10[#!]] &, n, 0] The plots can be displayed using the following Mathematica code, just changing the value of k: DiscretePlot[Floor[Sqrt[Log10[n!]]], {n, 0, k}, AxesLabel -> {"n", "a(n)=\!\(\*FormBox[TemplateBox[{\nSqrtBox[\nRowBox[{\n\ InterpretationBox[\nSubscriptBox[\"log\", \"10\"], Log10, \ AutoDelete -> True], \"(\", \nRowBox[{\"n\", \"!\"}], \")\"}]]},\n\ \"Floor\"], TraditionalForm]\)"}, LabelStyle -> Directive[Black, Bold]] Note: More detailed information regarding the process of reproducing the data can be located in the .nb file associated with this dataset.