| Version 13 |
Version 12 |
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We define the function $f : \mathbb{N} \setminus \{0\} \longrightarrow \mathbb{N} \setminus \{0\}$ such that
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We define the function $f : \mathbb{N} \longrightarrow \mathbb{N}$ so that
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| $$ f(n+1) = \left\{ |
$$ f(n+1) = \left\{ |
| \begin{array}{rl} |
\begin{array}{rl} |
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3f(n)+1 & \text{ if } f(n) \text{ is odd } \\
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3f(n)+1$ & f(n) \text{ is odd } \\
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f(n)/2 & \text{ if } f(n) \text{ is even.}
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f(n)/2$ & f(n) \text{ is even.}
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| \end{array} |
\end{array} |
| \right. $$ |
\right. $$ |
| $f(0)$ is an arbitrary seed value. |
$f(0)$ is an arbitrary seed value. |
| It is conjectured that the sequence $f(0),f(1),f(2),\ldots$ will always end in $1,4,2$, which repeats infinitely. This has been verified by computer up to very large values of $f(0)$, but is unproven in general. It is also not known whether this problem is decideable. |
It is conjectured that the sequence $f(0),f(1),f(2),\ldots$ will always end in $1,4,2$, which repeats infinitely. This has been verified by computer up to very large values of $f(0)$, but is unproven in general. It is also not known whether this problem is decideable. |
| This is sometimes called the ``hailstone sequence'' because, like a hailstone in a cloud, the values oscillate up and down. |
This is sometimes called the ``hailstone sequence'' because, like a hailstone in a cloud, the values oscillate up and down. |
