Thursday, January 1

Visitors from infinity

Sometimes a sequence in the OEIS doesn't do justice to the bigger picture. The blue trajectory, forward from the zero point to where it ends on the right is A251412. This integer sequence also has a backward history. Combined with an infinite number of like-minded sequences (four of which are shown; the identifying numbers at the forward end are the trajectories' minima at point zero) coming in from infinity at the left and going back out to it at the right, these trajectories meander for position in number space. Any outgoing trajectory running into an integer-point of an incoming one would of course merge with it. (Well, it was a single trajectory all along. We just didn't know it.) What if an outgoing trajectory were to run into its own tail? In that case, the trajectory is seen to be — not infinite — but finite. There are currently 34 known finite orbits in this mapping (which includes 7 fixed points, orbits of length 1). The currently longest orbit is one of length 91.

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