On a Primorial-Based Sequence and Its Primality Properties up to n = 1000

This record contains Version 2.1 of the research note “On a Primorial-Based Sequence and Its Prime-Generating Properties up to n = 1000”, together with links to the associated computational source code.

Changes in Version 2.1
- Added a brief explanation of the rationale for choosing the second prime strictly greater than each primorial.
- Refined the keywords to better reflect the scope and computational nature of the study.
- Renamed the research note file to better align with the terminology used in the paper, without changing its title.
- Made minor editorial adjustments to improve clarity and presentation.

Abstract
This work presents a computational exploration of the sequence defined by the displacement from the n-th primorial (P_n = p_n#) to the second prime greater than it (r_n = q₂ - P_n), for 1 ≤ n ≤ 1000. All computed terms in the tested range are prime, motivating a conjecture that this property holds for every positive integer n.

The paper establishes a restriction on the possible prime factors of r_n: no prime less than or equal to p_n can divide the term. From this result, a sufficient condition for primality is derived. The study also examines the empirical lower-boundary behavior of the computed sequence using a linear regression fitted to the minima of 25 consecutive blocks, each containing 40 terms. The observed frequency distribution of the output values is also analyzed.

These computational findings are limited to the evaluated range and do not establish the conjecture for arbitrary indices or determine the sequence's asymptotic behavior.

Files and Resources
- Research note (PDF).
- Link to the computational source code and data-generation repository.

Source Code and Repository
The source code and computational resources are publicly available on GitHub:
https://github.com/JCHernandezT/primorial-gap-recurrence