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Proceedings of the American Mathematical Society
Proceedings of the American Mathematical Society
ISSN 1088-6826(online) ISSN 0002-9939(print)

 

A new statistic for the $3x+1$ problem


Authors: David Gluck and Brian D. Taylor
Journal: Proc. Amer. Math. Soc. 130 (2002), 1293-1301
MSC (2000): Primary 11B83
Published electronically: November 9, 2001
MathSciNet review: 1879950
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Abstract: A finite $(3x+1)$-trajectory is a sequence $\underline{ a}=a_1,\ldots,a_n$ of positive integers such that $a_{i+1}=3a_i+ 1$ if $a_i$ is odd, $a_{i+1}=a_i/2$ if $a_i$ is even, $a_i>1$ if $i<n$ and $a_n=1$. For such a sequence $\underline{ a}$ we define $C(\underline{ a}) = (a_1a_2+\cdots+a_{n-1}a_n+a_na_1)/(a_1^2+\cdots+a_n^2)$. We prove that $9/13<C(\underline{ a})<5/7$ if $a_1$ is odd and $a_1\ge3$. Histograms suggest that $C$may have an interesting limiting distribution.


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Additional Information

David Gluck
Affiliation: Department of Mathematics, Wayne State University, Detroit, Michigan 48202
Email: dgluck@math.wayne.edu

Brian D. Taylor
Affiliation: Department of Mathematics, Wayne State University, Detroit, Michigan 48202
Email: bdt@math.wayne.edu

DOI: http://dx.doi.org/10.1090/S0002-9939-01-06305-5
PII: S 0002-9939(01)06305-5
Received by editor(s): November 7, 2000
Published electronically: November 9, 2001
Additional Notes: The first author’s research was partially supported by a grant from the National Security Agency
Communicated by: David E. Rohrlich
Article copyright: © Copyright 2001 American Mathematical Society