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. 1 updates: 1. Expanded Proposition 3. 1 to account for the parity dependence on the exponent 'a' in the relation Sₔ, ₕ, ₑ = 2ᵃ Mₙ. 2. Enhanced the formal proof of Propositon 3. 2 for single-term solutions by explicitly demonstrating why the exponent 'a' must be equal to 1, providing a more rigorous derivation. 3. Clarified the statistical analysis in the results section 3. 1 to better distinguish between the total population of solutions and the primary subset of solutions with 10 terms.
Juan Camilo Hernández-Torres (Fri,) studied this question.