Observations on Sums of Products of Prime Sequences and Mersenne Numbers

Experimental research note devoted to the computational analysis of sums of products of prime sequences under specific arithmetic constraints. The work investigates empirical regularities arising from corresponding prime sequences with fixed index displacement and their observed connections with Mersenne numbers and perfect numbers. The repository includes reproducible source code, computational datasets, statistical analyses, and visualizations associated with the exploratory results. Version 2.2 updates: 1. Corrected the explored upper bound for the parameter (r) in Section 3.3 from (2^26) to (2^27), ensuring consistency with the experimental setup described in Section 2.2. Corrected the exploration bound for the parameter (t_max) in Section 3.2 from (2^12) to (2^13) terms, and (2^14) to (2^16) for the parameter (u), ensuring consistency with the experimental configuration reported in Section 2.3. Improved the wording to preserve coherence with the subsequent concluding paragraph concerning the theoretical interpretation of the observed relations.4. Refined the terminology in Section 3.4 by replacing the expression “exact solutions” with a more concise formulation, improving stylistic consistency in the discussion of solutions associated with perfect numbers.