A new statistic for the $3x+1$ problem
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- by David Gluck and Brian D. Taylor PDF
- Proc. Amer. Math. Soc. 130 (2002), 1293-1301 Request permission
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 $a_1\ge 3$. Histograms suggest that $C$ may have an interesting limiting distribution.References
- Jeffrey C. Lagarias, The $3x+1$ problem and its generalizations, Amer. Math. Monthly 92 (1985), no. 1, 3–23. MR 777565, DOI 10.2307/2322189
- Riho Terras, A stopping time problem on the positive integers, Acta Arith. 30 (1976), no. 3, 241–252. MR 568274, DOI 10.4064/aa-30-3-241-252
- Günther J. Wirsching, The dynamical system generated by the $3n+1$ function, Lecture Notes in Mathematics, vol. 1681, Springer-Verlag, Berlin, 1998. MR 1612686, DOI 10.1007/BFb0095985
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
- 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
- © Copyright 2001 American Mathematical Society
- Journal: Proc. Amer. Math. Soc. 130 (2002), 1293-1301
- MSC (2000): Primary 11B83
- DOI: https://doi.org/10.1090/S0002-9939-01-06305-5
- MathSciNet review: 1879950