RFE June 2026: An iterated divisibility construction

[Sean A. Irvine]

Hi,

Recently I came across an "uned" sequence which looks interesting:

The "uned" sequence A140595 (Paul Curtz) is based on the diagram in A140485 (Eric Angelini) which is in turn an iterative divisibility construction.

Is A140595 well-defined? Can someone please compute more terms?

I track these requests for enhancement here:

https://oeis.org/wiki/User:Sean_A._Irvine/Requests_for_Enhancements#Requests_for_Enhancements

Sean.

[Thomas Scheuerle]

It is well defined if we know that each column in the diagram from A140485 has finite length.

Also we should be able to know if we have reached all numbers in a column to be able to calculate this sequence.

The maximum possible length of such a column is essentially limited by the maximum of the first differences in A140485.

Obviously this maximum would not have any bound if A140485 could contain numbers like factorials or primorials etc... , 

so we need to investigate what structure of divisors is possible for the members of A140485.

[Misha Lavrov]

Even the diagram is not necessarily well-defined (before we get to any question of whether the columns are finite or infinite) if it is possible that two numbers have disjoint trajectories under the "n -> n + second-smallest number that does not divide n" map. This seems incredibly unlikely, but if it does happen, then the diagram has multiple components with no natural way of aligning columns in different components.

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[Allan Wechsler]

I think the following has some points in common with Thomas Scheuerle; we were typing at the same time. But I have some things he doesn't. Here is my contribution.

Oeis.org/A140485 is "trajectory of 1 under n -> n + second smallest number that does not divide n".

My usual first instinct here is to make sure that all obviously simpler concepts have their own entry. In this case, I first stripped off "trajectory of 1 under ..." and went to see if "n + second smallest number that does not divide n" was archived. I read the numbers off the tree in the Example section, and got 4,6,7,9,8,11,10,13,13,..., which is not in OEIS.

Then I stripped off the "n +", and went to see if "second smallest number that does not divide n" was recorded. I think this sequence would be 3,4,4,5,3,5,3,5,4,..., which is also not in the OEIS. 

The tree in the Example section of A140485 is indeed very striking. It certainly looks like each column contains a contiguous set of integers.

The following conjecture must be true for oeis.org/A140595 to be well-defined: the trajectory of any integer k under the map "n -> n + second smallest integer that does not divide n" eventually joins the trajectory of 1 under that map. So a sequence simpler than A140595 with the same well-definedness condition would be "The number of iterations of this map, starting from n, to join the trajectory of 1". If we added "-1 if it never joins", the sequence would be well-defined already. But I think the conjecture is true: the "Angelini increment" is usually pretty small (see my second proposed sequence above), so the trajectories grow slowly and have ample opportunity to interact with each other. I suspect this may even be provable without too much difficulty.

Now I see another post has beat mine, so I'd better post this now, in hopes somebody else has more insight.

The ghost of Éric Angelini continues to entertain us!

-- Allan

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[Thomas Scheuerle]

If A140485 can not contain any multiple of 120, then columns will not be longer than 5 and can also not be disjoint, right?

[Allan Wechsler]

I misinterpreted Éric's tree, and Thomas has set me straight. My wrong impression was that each number was positioned so that one application of the successor-map would advance one column. But this isn't always true: 30 is shown in column 8, and its successor, 37, is in column 10. So now I don't know what the construction principle of the tree is. I think that the consecutivity of the elements of the columns is a principle of construction, but I can't give a coherent description of the tree as a whole.

Proposed nomenclature for this discussion: the Angelini increment of n is the second smallest nondivisor of n; the Angelini successor of n is n incremented by its Angelini increment.

I think that the machinery Clark Kimberling calls a dispersion array may be relevant here.

-- Allan

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[Thomas Scheuerle]

If no multiple of 120 is reached, we know that the sequence must become periodic under A140485(n) mod c, where c = k*120 with some small constant k.

And if it has becomes periodic, then this will further on exclude any multiples of 120.

By simply looking on the data by eye ( no computer today ), it seems that this is already the case.

If true, no multiples of 120 and such a limited length of columns.

The remaining problem is the exact rule how to order the numbers. I think Éric had not in mind that its tree should be a sequence,

he used an ordering that did fit good into a diagram for visualization, so the definition of A140595 regarding the order how the numbers will appear seems to be a bit convoluted and questionable now.