Conway's alimentary function (A070871)

[William Orrick]

I have a draft edit of this sequence, currently in the queue, and am trying to find out more about it. This is a fairly natural sequence as it is tied to fundamental properties of Stern's diatomic series, the Stern-Brocot tree, the Calkin-Wilf tree, and the modular group. In particular, the difference between the rational numbers labeling adjacent regions in the Stern-Brocot tree is always the reciprocal of an integer, and it is natural to label the edges of the tree with that integer, that is, with the terms of A070871.

The sequence, along with a number of companion sequences, was added to the OEIS in 2002. Most of the companion sequences seem to be tied to an exploration of what happens when you iterate the function. In particular, A071887 is a list of values for which the iterated function is conjectured not to reach a cycle.

I have been unable to find any information about this, apart from what's in the OEIS. My questions are:

1) Is there any literature on this sequence?

2)  Does anyone know what Conway's motivation was for looking at iterates of this function? The indexing of A070871 is at least slightly arbitrary. Shifting the offset would produce different cycles and trajectories.

3) What is the reason for the term "alimentary function" and the term "nutritionally balanced numbers" for A070872?

4) Sean Irvine left a comment on A071887 in 2011 pointing out that the value 42 appears to be missing, as it doesn't seem to lead to a cycle. In fact, 42, 43, 52, and 53 all lead to the same trajectory. So if 42 doesn't lead to a cycle, that would be four missing values between 41 and 54. The next value after 58 that doesn't lead to to an obvious cycle is 65, which leads to the same trajectory as 42. After that, the next conjectural value would be 68, which leads to the same trajectory as 38. Of the six values given in the present version of the sequence, 37 and 58 lead to the same trajectory, 38 and 57 lead to the same trajectory, and 41 and 54 lead to the same trajectory. This is expected due to a symmetry of A070871, regarded as an array. Can anyone confirm or extend any of these calculations or, better, prove any of these results? (Assuming there's a good motivation for putting time and effort into this.) Up to n = 200 there seem to be 17 infinite trajectories and only a handful of cycles, but my iterations don't extend very far, so if there are long cycles, I've missed them.

Best,

Will

[William Orrick]

Just an update. I have computed the trajectory of 42 out to the 114th iterate and observed very rapid growth. The 114th iterate is more than 2 million digits long; the number of digits appears to double roughly every 5 or 6 iterates. This is consistent with the comments left by Sean A. Irvine and Charles R. Greenhouse IV back in 2011:

Irvine: Why isn't 42 in this list? It does not enter a cycle in under 100 iterations which is as far as I could check.

Irvine: Perhaps this sequence would better list number of steps to enter a cycle, or 0 for numbers with unknown status.

Greathouse: 43, 52, 53, 65, ... are in the same trajectory as 42. I agree, it looks like these do not cycle -- in fact the values are growing quite rapidly. Look at the binary values, too -- they're just what you'd expect for a sequence that will grow under fusc.

I haven't investigated the binary digits yet. Perhaps there is some pattern that explains the name "alimentary function" and the interest in iterating the function.

I do tend to think that 42 does not enter a cycle. On the other hand, T. D. Noe wrote, "Perhaps Conway has another way to determine these numbers." However, if there's some hidden structure that causes the rapid growth to stop eventually I suspect a mention of that would have been made in one of the existing sequences submitted at the same time as A071887. Instead, at A071884 we see, "All smaller starting values lead into a cycle after a small number of steps; this appears to have an infinite trajectory" along with some fairly standard code for computing the sequence. If there is a surprise long-run behavior that needs to be considered, I would think it would have been mentioned there.

So I propose, in the absence of any information to the contrary, that 42, 43, 52, 53, 65, 68, and perhaps a few more terms be included in sequence A071887.

[William Orrick]

For those who are keeping score at home, there is an error in my previous comment: the trajectory of 42 first exceeds two million digits on the 115th iterate, not the114th. (This is counting 42 itself as the zeroth iterate.) I stopped the calculation at the 120th iterate, which has about 3.9 million digits. It could be continued beyond that, but I don't see much point.

Based on further calculations, I now believe cycles to be the exception and infinite trajectories to be the rule. Of the first five million natural numbers, fewer than 2000 are known to lead to a cycle. (For each natural number I stop the calculation when an iterate exceeds 10^20.) I only find ten cycles, seven of length 1, one of length 2, and two of length 3. Only the cycle [2] and the cycle [36, 44, 60] are reached with any frequency. For all I know, it could be that there are only finitely many natural numbers that lead to a cycle. Can anyone prove otherwise?

I notice that the comment at A070837 is incorrect. The comment states that it appears that if n is in the sequence, then so is 2n. But the sequence contains 20 but not 40,  36 but not 72, and 44 but not 88. I have confirmed that the sequence is correct, so the claim is genuinely false. I know that one is allowed to correct comments, but I can't think of any correction to make here. Can the comment simply be deleted?

-Will

[Sean A. Irvine]

> Can the comment simply be deleted?

Yes. Genuinely incorrect information can be proposed for deletion. Use an extension like "Incorrect comment removed by".

Sean.

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