The 3x+3<sup>k</sup> generalised Collatz mapping

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The $3x+3^k$ generalised Collatz mapping

Consider the mapping T_k: 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
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The 3x+3k3x+3^k generalised Collatz mapping

The $3x+3^k$ generalised Collatz mapping