A combinatorial partition of Mersenne numbers arising from spectroscopy

An analytic proof has been developed for a conjecture postulating a finite and non-integer, though rational, partitioning of Mersenne numbers, i.e., those of form 2n−1. This conjecture had arisen from a pattern of values observed for interference coefficients present in a system of absorbers in transmission spectroscopy. A set of foundational lemmas is presented and proven as an aid in simplifying the proof of the conjecture. The proof also establishes the validity of related partitions for various number families, such as even perfect numbers, in which a Mersenne number type of factor is present.