Extended dataset produced by the Integer sequence (A363743)
Author
Paul F. Marrero Romero
Title
Extended dataset produced by the Integer sequence (A363743)
Description
Numerical Data produced by the sequence A363743, for integer values of n, in the range of: 0 <= n <= 5000.
Category
Academic Articles & Supplements
Keywords
mathematics, mathematica, sequences, integer sequences, oeis, discrete mathematics, number theory, code, dataset
URL
http://www.notebookarchive.org/2023-12-2ecqz2i/
DOI
https://notebookarchive.org/2023-12-2ecqz2i
Date Added
2023-12-05
Date Last Modified
2023-12-05
File Size
358.56 kilobytes
Supplements
Rights
CC BY 4.0
Download
Open in Wolfram Cloud
Extended dataset produced by the Integer sequence (A363743): a(n) = log10(n!)
Extended dataset produced by the Integer sequence (A363743): a(n) =
(n!)
log
10
Paul F. Marrero Romero
Numerical Data produced by the sequence a(n) = , for integer values of n, in the range of: 0 ≤ n ≤ 5000.
(n!)
log
10
Details
Details
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, but mathematically it corresponds to a(n) = . On the other hand, it was possible to represent the general formula on another two forms that the following: a(n) = floor(sqrt(A034886(n) - 1)).2) a(n) = A000196(A034886(n) -1).A034886 on the OEIS represents the integer sequence that produces the “number of digits in n!” Additionally, A000196 is a sequence that represents “the integer part of the square root of n,” “number of positive squares <= n,” or “n appears 2n+1 times” in the OEIS database. Some interesting properties in A363743 are:
(n!)
log
10
1)
◼
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 for any non-negative integer m.
a(n)
a(n+m)
About the Mathematica code utilized for reproducing the data on the On-Line Encyclopedia of Integer Sequences, we utilized the following code:
Array[Floor@ Sqrt[Log10[#!]] &, 93,0]
Array[Floor@ Sqrt[Log10[#!]] &, 93,0]
The previous code executed in 0.012913 seconds (Wolfram Cloud) and 0.015625 seconds (Personal Computer).
The execution times were measured using the Mathematica function “Timing[]” i.e.,
The execution times were measured using the Mathematica function “Timing[]” i.e.,
Timing[Array[Floor@Sqrt[Log10[#!]]&,93,0]]
The dataset reported on this notebook was generated by the following Mathematica program:
Array[Floor@Sqrt[Log10[#!]]&,5000,0]
Which has an execution time of: 0,823717s (Wolfram Cloud) and 1,28125s (Personal Computer).
Other program used to generate the data reported in A363743 is the following PARI - based program, courtesy of Michel Marcus (OEIS Staff Member):
Other program used to generate the data reported in A363743 is the following PARI - based program, courtesy of Michel Marcus (OEIS Staff Member):
(PARI)a(n)=sqrtint(log(n!)/log(10));
Below there are two graphs displaying the sequence behavior as reported on the Encyclopedia, utilizing the same amount of data:
Data Definitions
Data Definitions
Mathematica code that computes the sequence of integers a(n) = log10(n!) for non-negative integers n such that n ≤ 5000.
Mathematica code that computes the sequence of integers a(n) = for non-negative integers such that n ≤ 5000.
(n!)
log
10
n
In[]:=
Array[Floor@Sqrt[Log10[#!]]&,5000,0](*Basecode*)
In[]:=
DS=Array[Floor@Sqrt[Log10[#!]]&,5000,0](*Dataset*)
Mathematica code demonstrating the behavior of the sequence when evaluated for a nonnegative integer n, where n is less than or equal to 5000:
Mathematica code demonstrating the behavior of the sequence when evaluated for a nonnegative integer n, where n is less than or equal to 5000:
In[]:=
DiscretePlotFloor[Sqrt[Log10[n!]]],{n,0,4999},AxesLabel->"n","a(n)=(n!)",LabelStyle->Directive[Black,Bold]
log
10
In[]:=
PT1=DiscretePlotFloor[Sqrt[Log10[n!]]],{n,0,4999},AxesLabel->"n","a(n)=(n!)",LabelStyle->Directive[Black,Bold]
log
10
Primary Content
Primary Content
Warning: To accurately reproduce the following Data and Plot, execute all Input code lines stated in the Data Definitions prior.
In[]:=
DS
In[]:=
PT1
Out[]=
Examples
Examples
Using Mathematica we have that:
In[]:=
a[7]=Floor[Sqrt[Log10[7!]]](*a(7)=1*)
Out[]=
1
In[]:=
a[431]=Floor[Sqrt[Log10[431!]]](*a(431)=30*)
Out[]=
30
In[]:=
a[3973]=Floor[Sqrt[Log10[3973!]]](*a(3973)=112*)
Out[]=
112
Source & Additional Information
Source & Additional Information
Submitted By
Submitted By
Paul F. Marrero Romero
Source/Reference Citation
Source/Reference Citation
Paul F. Marrero Romero, a(n) = floor(sqrt(log_10(n!))). , Entry A363743 in The On-Line Encyclopedia of Integer Sequences, https://oeis.org/A363743
Detailed Source Information
Detailed Source Information
Author/Creator
Author/Creator
Paul F. Marrero Romero
Source Title
Source Title
a(n) = floor(sqrt(log_10(n!))).
Source Date
Source Date
August 17 - 2023.
Source Publisher
Source Publisher
Geographic Coverage
Geographic Coverage
Worldwide
Source Language
Source Language
English
Links
Links
◼
Integer sequence: A000196
◼
Integer sequence: A034886
Keywords
Keywords
◼
Sequences
◼
Mathematica
◼
Integer sequences
◼
OEIS
◼
Discrete Mathematics
◼
Factorial
◼
Analysis
◼
Algorithm
Categories
Categories
Content Types
Content Types
Author Notes
Author Notes
This sequence is currently under my study, and it is my intention to produce a paper discussing its algebraic properties and other aspects after proof of the corresponding mathematics. The sequence is the result of a personal research project that I am conducting in the field of discrete mathematics.
Personal Computer specs:
◼
Intel(R) Core(TM) i3-4005U CPU @ 1.70GHz 1.70 GHz
◼
6,00 GB Ram - DDR4
◼
SO: Windows 10 x64 Professional.
Cite this as: Paul F. Marrero Romero, "Extended dataset produced by the Integer sequence (A363743)" from the Notebook Archive (2023), https://notebookarchive.org/2023-12-2ecqz2i
Download
