Extended Dataset Generated by the OEIS Integer Sequence A377045: Number of Partitions of Cuban Primes.

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.

此整数序列于2024年10月14日在整数序列在线百科全书（OEIS.org）数据库中注册并公开发布，其OEIS代码为：A377045。该序列可通过以下两个通用公式表达，这两个公式涉及以下序列：

1) A000041：a(n)代表n的划分数（划分数）。

2) A002407：古巴素数：两个连续立方数的差为素数的素数。

3) A121259：满足(3*k^2 + 1)/4为素数的整数k。

上述两个通用公式如下：

a(n) = A000041(A002407(n))。 (1)

a(n) = A000041((3*A121259 (n)^2+1) / 4)。 (2)

该序列的一些有趣性质包括：

◼ 两个连续立方数差为素数的素数的划分数。

◼ 满足p=(3*n^2 + 1) / 4（对于某些整数n）的素数p的划分数（A121259）。

◼ a(13) ≈ 1.49910 × 10^43。

◼ 已知的A121259中的最后一个整数n为341，对应于a(60) ≈ 1.59114 × 10^323。

该数据集中的数值数据由以下Mathematica程序生成：

PartitionsP[Select[Table[(3 k^2 + 1)/4, {k, 500}], PrimeQ]]

前述程序基于Mathematica v13.3.0构建。

注：更多数学细节、图形和技术信息可在本数据包提供的笔记本(.nb)及PDF文件中找到。