Journal of Integer Sequences, Vol. 14 (2011), Article 11.2.3

A Function Related to the Rumor Sequence Conjecture

Bruce Dearden, Joel Iiams, and Jerry Metzger
University of North Dakota
Department of Mathematics
Witmer Hall Room 313
101 Cornell Street Stop 8376
Grand Forks, ND 58202-8376


For an integer $ b\geq 2$ and for $ x\in [0,1)$, define $ \rho_b(x) = \sum_{n=0}^{\infty} \frac{\{\mskip -5mu\{b^nx\}\mskip -5mu\}}{b^n}$, where $ \{\mskip -5mu\{t\}\mskip -5mu\}$ denotes the fractional part of the real number $ t$. A number of properties of $ \rho_b$ are derived, and then a connection between $ \rho_b$ and the rumor conjecture is established. To form a rumor sequence $ \{z_n\}$, first select integers $ b\geq 2$ and $ k\geq 1$. Then select an integer $ z_0$, and for $ n\geq 1$ let $ z_n = bz_{n-1} \bmod{(n+k)}$, where the right side is the least non-negative residue of $ bz_{n-1}$ modulo $ n+k$. The rumor sequence conjecture asserts that all such rumor sequences are eventually 0. A condition on $ \rho_b$ is shown to be equivalent to the rumor conjecture.

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(Concerned with sequences A049802 A049803 A080277 A080333 A177356 .)

Received December 1 2010; revised version received February 1 2011. Published in Journal of Integer Sequences, February 19 2011.

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