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3 results for “Integer sequences”
Comparative plot about two sequences of integers very similar to each other: A348960 vs A127034
<p>This is the behavior comparison graph between the integer sequences registered in OEIS (The On-Line Encyclopedia of Integer Sequences) with codes: A348960 & A127034 respectively.</p> <p><strong>1)</strong> The sequence A348960 obeys the formula: </p> <p> <span class="math-tex">\(a_{n }=\lfloor(log(\pi n!)\rfloor. \)</span> For any non-negative integer such that <span class="math-tex">\(n\geq0\)</span></p> <p><strong>2) </strong>The sequence A127034 obeys the formula:</p> <p><span class="math-tex">\(a_{n}=\lfloor log(n!)/log(11)\rfloor.\)</span> For any non-negative integer such that <span class="math-tex">\(n\geq0\)</span></p> <p>In this particular case we're conducting the study for the following n-values: i<span class="math-tex">\(1\leq n\leq 60.\)</span> </p> <p> </p>
THE INTEGER SEQUENCE A348960
<p>This is a sequence of integers that was registered by me in the On-line Encyclopedia of Integers (OEIS) under the code: A348960. In this report, the sequence is evaluated for integers <strong>n,</strong> taking values within the range [0,70]. An algorithm in MATHEMATICA for the sequence under study is also presented.</p> <p>The empirically discovered sequence is as follows: <span class="math-tex">\(a(n) = \lfloor log(\pi n!)\rfloor\)</span></p> <p> </p>
Extended dataset and Mathematica notebook produced by the Integer sequence (A363743): a(n) = floor(sqrt(log_10(n!))).
<p>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. </p><p>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: </p><p>1) a(n) = floor(sqrt(A034886(n) - 1)).</p><p> 2) a(n) = A000196(A034886(n) -1). </p><p>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: </p><p>* Every non-negative integer occurs at least 4-times. </p><p>* Each integer k > 14 appears fewer than k times. </p><p>* The only integers k that occur exactly k times are 11, 13 and 14. </p><p>* This sequence can produce random values between 0 and 1 if we do a(n)/a(n+m) for any non-negative integer m. </p><p>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.</p><p> <strong>Note:</strong> More mathematical details, graphics and technical information can be found in the notebook or .nb file provided in this dataset. </p>
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