2020/08/22 by Anant Gupta, Gupta, Anant
Computer Science · Mathematics · #11D45 #Benford’s Law and Fraud Detection #Discrete Mathematics (cs.DM) #FOS: Computer and information sciences #FOS: Mathematics #Number Theory (math.NT) #cs.DM #math.NT #msc:11D45
paper · pdf · doi:10.48550/arxiv.2008.11103
23 pages, 10 figures, 6 tables
arxiv created 2020/08/22 · openalex publication_date 2020/08/22 · arxiv updated 2020/08/26 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We explore the cycles and convergence of Generalized Collatz Sequence, where 3n+1 in original collatz function is replaced with 3n+k. We present a generating function for cycles of GCS and show a particular inheritance structure of cycles across such sequences. The cycle structure is invariant across such inheritance and appears more fundamental than cycle elements. A consequence is that there can be arbitrarily large number of cycles in some sequences. GCS can also be seen as an integer space partition function and such partitions along with collatz graphs are inherited across sequences. An interesting connection between cycles of GCS and certain exponential Diophantine equations is also presented.