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Preferred Values of Collatz Paths

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size
75
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Consider the function
n↦
n
2
niseven
3n+1
nisodd.
The Collatz problem consists in repeatedly evaluating the function for every positive integer
n
; it always seems to end in 1, but this may not always be true. This shows a plot of paths for the Collatz problem; each path is plotted along the
y
axis. Notice that the plot reveals a pattern of diagonal lines that pass though the origin and horizontal lines that show that certain values are much more likely than others. The reason for these diffuse patterns made by preferred values is unclear.
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