On a Recursive Prime-Factor Tree Representation for Partial Sums of Integer Sequences

This record contains Version 1.0 of the research note “On a Recursive Prime-Factor Tree Representation for Partial Sums of Integer Sequences”, together with a link to the associated Python implementation and computational validation scripts. AbstractThis note introduces a recursive prime-factor tree representation for finite sequences of integers and their partial sums. The construction is based on the smallest prime factor (SPF) and organizes integer terms into a hierarchical structure of raw and scaled nodes. The framework accommodates positive integers, negative integers, and zero, while preserving the multiplicities and signs of the input terms. The equivalence between a global multiset-based definition and an incremental insertion procedure is established. The resulting representation is shown to preserve the algebraic sum of the input terms and to be invariant under permutations of the input sequence. Furthermore, its canonical structure is characterized, and an upper bound on its depth is derived in terms of the total number of prime factors of the input integers. Illustrative examples and computational experiments are provided to examine the construction and validate its mathematical properties across exhaustive and randomized test cases. These experiments complement the theoretical results and provide computational evidence of the implementation's consistency. Files and Resources- Research note (PDF).- Link to the associated Python implementation and computational validation repository. Source Code and RepositoryThe Python implementation and computational validation scripts are publicly available on GitHub:https://github.com/JCHernandezT/prime-factor-tree-representation