Extended Dataset Generated by the OEIS Integer Sequence A376895: Primes of the Form 3^k*k^3 + 2.
Description
This integer sequence was registered and published in the On-Line Encyclopedia of Integer Sequences (OEIS.org) Database on October 08 - 2024, under the OEIS code: A376895. This sequence can be expressed with the help of a general formula that uses the OEIS-indexed sequence A366997 which are the integers n such that a(n) = 3n *n3 + 2 is a prime number. This general formula is such that: a(n) = 3^A366997 * A366997^3 + 2. Some interesting properties of this sequence are: ◼ The next term a(8) = ∼2.20847 x10143 is too large to include in the standard OEIS format. ◼ a(9) = ∼8.66244 x10153. ◼ a(10) = ∼9.21872 x10433. ◼ This sequence is constituted by primes of the forms A002145 and A002144, with the exception of 2. ◼ The last known integer k in A366997 is 20803 and corresponds to a(13) = ∼3.21988 x109938. The numerical data showed on this dataset was generated by the following Mathematica program: Select[Table[3^k*k^3+2, {k, 0, 10000}], PrimeQ] The previous program was builded on Mathematica v13.3.0. Note: More mathematical details, graphics and technical information can be found in the notebook (.nb) & pdf files provided in this data pack.
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Steps to reproduce
All the numerical data obtained in this dataset, which consists of terms that are prime numbers of the form a(n) = 3^k * k^3 + 2, where k is an integer such that k >=0, were generated using a Mathematica program that differs only in the search domain from the Mathematica program published in the On-Line Encyclopedia of Integer Sequencces (OEIS). Steps to reproduce the dataset: 1) Run the next Mathematica code on a Wolfram Language environment: Select[Table[3^k*k^3+2, {k, 0, 10000}], PrimeQ] 2) To get a plot related to the same amount of data reported in the OEIS database of this sequence, just run the next Mathematica code in a Wolfram Language environment: ListPlotSelectTable[3^k * k^3 + 2, {k, 0, 170}], PrimeQ, PlotLabel → HoldForm[A376985], Joined → True, AxesLabel -> "n", "p = 3^n * n^3 + 2 " , LabelStyle -> Directive[Black, Bold] 3) The calculation of the term "a(13)" of the sequence was realised using the last known integer k in the OEIS integer sequence A366997 (Numbers m such that 3^k*k^3 + 2 is a prime.), which is k =20803, together with a Mathematica code line that emulates the general formula corresponding to this integer sequence. To get this a(13), just run the following Mathematica code line in a Wolfram Language environment: 3^20 803 * 20 803^3 + 2 // N Note: Other examples and details about the extended dataset of A376895 and general information were included in the Mathematica notebook file in this data pack. Acknowledgements: All these analyses and calculations were carried out in the Marrero Reseach lab.