Extended Dataset Generated by the OEIS Integer Sequence A377045: Number of Partitions of Cuban Primes.
Description
This integer sequence was registered and published in the On-Line Encyclopedia of Integer Sequences (OEIS.org) Database on October 14 - 2024, under the OEIS code: A377045. This sequence can be expressed with the help of two general formulas that uses the sequences: 1) A000041: a(n) is the number of partitions of n (the partition numbers). 2) A002407: Cuban primes: primes which are the difference of two consecutive cubes. 3) A121259: Numbers k such that (3*k^2 + 1)/4 is prime. The two aforementioned general formulas are as follows: a(n) = A000041(A002407(n)). (1) a(n) = A000041((3*A121259 (n)^2+1) / 4). (2) Some interesting properties of this sequence are: ◼ Number of partitions of prime numbers that are the difference of two consecutive cubes. ◼ Number of partitions of primes p such that p=(3*n^2 + 1) / 4 for some integer n (A121259). ◼ a(13) = ~1.49910(x10^43). ◼ The last known integer n in A121259 is 341 and corresponds to a(60) = ~1.59114(x10^323). The numerical data showed on this dataset was generated by the following Mathematica program: PartitionsP[Select[Table[(3 k^2 + 1)/4, {k, 500}], 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.
Files
Steps to reproduce
All the numerical data obtained in this dataset, which consists of terms which represents the number of partitions of the cuban primes, or primes of the form p = (3*k^2 + 1) / 4 such that k is a non-negative integer, 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 Sequences (OEIS). -> Steps to reproduce the data: 1) Execute the next Mathematica code on a Wolfram Language console or environment: PartitionsP[Select[Table[(3k^2+1)/4,{k,500}],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 console or environment: ListPlot[PartitionsP[Select[Table[(3 k^2 + 1)/4, {k, 50}], PrimeQ]], PlotLabel -> HoldForm[A377045], Joined -> True, AxesLabel -> {"n", "\!\(\*FormBox[\(a(n)\), TraditionalForm]\)"}, LabelStyle -> Directive[Black, Bold]] 3) The calculation of the term "a(60)" of the sequence was realised using the last known integer k in the OEIS integer sequence A121259 (Numbers k such that (3*k^2 + 1)/4 is prime), which is k =341, together with a Mathematica code line that emulates the general formula (2) corresponding to this integer sequence. To get this a(60), just run the following Mathematica code line in a Wolfram Language console or environment: PartitionsP[(3*341^2 + 1) / 4]// N Note: Further examples and details of the extended dataset of A377045 and general information have been included in the Mathematica notebook file in this data package. Acknowledgements: All these analyses and calculations were carried out in the Marrero Research laboratory.