The \(3x+3^k\) generalised Collatz mapping
Consider the mapping Tk: ℤ → ℤ
where k ≥ 0:
\[
T_k(x)=\left\{\begin{array}{cl}
x/2& \mbox{ if \(x\) is even,}\\
(3x+3^k)/2& \mbox{ if \(x\)is odd.}
\end{array}
\right.
\]
In an email dated 2nd February 2001, Willem Maat observed that the sequence of iterates \(x, T_k(x), T_k^2(x),\ldots\) eventually becomes \(3^km\) for some odd integer \(m\) and
subsequently the iterates become \(3^km, 3^kT_0(m)\), \(3^kT^2_0(m),\ldots\), where \(T_0\) is the Collatx \(3x+1\) mapping.
Conseqently one expects to reach one of the cycles defined by \(0, 3^k, -3^k, -5\times 3^k, -17\times 3^k\).
This was also recently communicated to me on 31st October 2024 by David Barina.
There is a BC version available.
Last modified 7th November 2024
Return to main page