Attractors and basins generated by repeated sums of prime factors of natural numbers
Abstract
Integer maps are discrete dynamical systems defined on the natural numbers. In this paper, we investigate the dynamics of the shifted Alladi-Erd{\H o}s map, a one-parameter family of integer maps in which each composite number is mapped to the sum of its prime factors, while each prime is mapped to n+A, where A ∈ ℕ is a fixed shift parameter. By systematically exploring the parameter space for 2 ≤ A ≤ 105, we uncover a rich bifurcation structure characterized by the emergence, disappearance, and reorganization of attractor cycles as the shift parameter varies. We find that although several attractors may coexist for a given value of A, for most values of A, almost all natural numbers belong to the basins of only two dominant attractors. Using elementary number theoretic arguments, we explain the observed bifurcation diagram. We further characterize the attractor cycles and quantify the distribution of their basin sizes across the parameter space. Our results reveal an unexpectedly rich landscape of arithmetic dynamics arising from a remarkably simple integer map. %and provide a systematic characterization of how attractors and their basins evolve as shift parameter is varied.
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BibTeX
@misc{shekatkar2026attractors,
title = {{Attractors and basins generated by repeated sums of prime factors of natural numbers}},
author = {Snehal Shekatkar and Ivano Lodato},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.29232},
eprint = {2609.29232},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.29232},
note = {CC BY-NC-ND 4.0},
}
RIS
TY - UNPB
TI - Attractors and basins generated by repeated sums of prime factors of natural numbers
AU - Snehal Shekatkar
AU - Ivano Lodato
PY - 2026
DA - 2026/09/24
PB - arXiv
DO - 10.48550/arxiv.2609.29232
UR - https://arxiv.org/abs/2609.29232
AB - Integer maps are discrete dynamical systems defined on the natural numbers. In this paper, we investigate the dynamics of the shifted Alladi-Erd{\H o}s map, a one-parameter family of integer maps in which each composite number is mapped to the sum of its prime factors, while each prime is mapped to n+A, where A ∈ ℕ is a fixed shift parameter. By systematically exploring the parameter space for 2 ≤ A ≤ 10^5, we uncover a rich bifurcation structure characterized by the emergence, disappearance, and reorganization of attractor cycles as the shift parameter varies. We find that although several attractors may coexist for a given value of A, for most values of A, almost all natural numbers belong to the basins of only two dominant attractors. Using elementary number theoretic arguments, we explain the observed bifurcation diagram. We further characterize the attractor cycles and quantify the distribution of their basin sizes across the parameter space. Our results reveal an unexpectedly rich landscape of arithmetic dynamics arising from a remarkably simple integer map. %and provide a systematic characterization of how attractors and their basins evolve as shift parameter is varied.
ER -