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Generalized Collatz Functions: Cycle Lengths and Statistics

Authors:

Hayden R. Messerman ,

Central Washington University, US
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Joey LeBeau,

Central Washington University, US
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Dominic Klyve

Central Washington University, US
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Abstract

Consider the function T (n) defined on the positive integers as follows. If n is even, T (n) = n/2. If n is odd, T (n) = 3n + 1. The Collatz Conjecture states that for any integer n, the sequence n, T (n), T (T (n)), . . . will eventually reach 1. We consider several generalizations of this function, focusing on functions which replace "3n + 1" with "3n + b" for odd b. We show that for all odd b < 400, and all integers n ≤ 106, iterating this function always results in a finite cycle of values. Furthermore, we empirically observe several interesting patterns in the lengths of these cycles for several classes of values of b.

Faculty Sponsor: Dr. Dominic Klyve

How to Cite: Messerman, H.R., LeBeau, J. and Klyve, D., 2012. Generalized Collatz Functions: Cycle Lengths and Statistics. International Journal of Undergraduate Research and Creative Activities, 4(1), p.2. DOI: http://doi.org/10.7710/2168-0620.1002
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Published on 09 Oct 2012.
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