Aliquot sequences that start on the integer powers n^i

[richs]

392^55 terminates P3 at i94.

[garambois]

Originally Posted by yoyo

I go a base up to n^100 and a sequence up to composites of C139.
So e.g. for base 288 288^61, 288^67 and 288^68 are handled right now.

When I take a base, I take everything from n^1 to n^100, which has not finished and has a composite < C140.

I understand that your criteria make you handle sequences that are not taken into account by our project page.
If you handle base 288 with exponents > 60, it raises the question for me :

Do you think I should extend all bases < 1000 to exponent 70 on the project page ?
Please give me your opinions on this question, because the limit exponents I chose are not a rigorous choice.

For example for base 999, we have 999^70 which has 210 digits !
It's just to avoid nonsense.
I can't put 8128^100 on the page, because that number has 391 digits !

[garambois]

Originally Posted by sweety439

Conjectures:

* If n is odd, then 276^n never terminate.
* If n is even, then 276^n must terminate.

This happens for all bases.

Reminder :
- Sequences whose bases and exponents have the same parity usually terminate trivially (the only exception currently known : 29^15).
- Sequences whose bases and exponents do not have the same parity end very rarely.
- For bases that are doubles of squares, all sequences end trivially in general (no exception known at the moment).

This is due to a theorem that says that in a sequence, there is a change of parity only when a term is a perfect square or the double of a perfect square.
And in general, when we have odd terms in a sequence, it ends quickly.

[garambois]

Originally Posted by Happy5214

IMO bases which themselves are main project sequences (like the Lehmer five) should form a new category on the main page given their particular notability.

Honestly, I am against this idea.
Indeed, if Catalan's conjecture is true, then the sequences of the main project will all come out of the main project one day.
The placement of a base in this hypothetical new category would therefore be in no way final, unlike the placements of bases in all our other current categories.
But I agree that creating a category for the Open-End bases of the main project would facilitate the statistics !
Because, in the other hand, it is very interesting indeed to calculate the sequences of the bases in the main project.
I had already noticed that for the Lehmer five, non-trivial sequences ending seemed to be very rare.
But I think this must be a coincidence.
Otherwise, we would have a very curious conjecture !
It's a great idea to dig into this question !

[garambois]

Originally Posted by sweety439

The page has "Primorials", but does not have "Factorials", I try to take the factorials.

Also, I try the highly abundant numbers, since they are the numbers whose sigma function sets record, and sigma function is highly related to Aliquot sequences.

Besides, there are also interesting bases: 102 and 138, see https://oeis.org/A098009, they set record for the length of Aliquot sequences.

Finally, not only the Lehmer five, there are also other numbers less than 1000 which is conjectured to have an infinite, aperiodic, aliquot sequence: 306, 396, 696, 780, 828, 888, 996, which have the same trajectories as the Lehmer five.

Let me know if you manage to initialize bases with the minimum rules we set : for a given base, compute all sequences <120 digits that end trivially and <100 digits that are Open-End.
These are only the minimal rules, you can do more if you have big computational resources ;-)

[sweety439]

Originally Posted by garambois

Let me know if you manage to initialize bases with the minimum rules we set : for a given base, compute all sequences <120 digits that end trivially and <100 digits that are Open-End.
These are only the minimal rules, you can do more if you have big computational resources ;-)

OK, reserve base 120 (= 5!)

[yoyo]

Originally Posted by garambois

I understand that your criteria make you handle sequences that are not taken into account by our project page.
If you handle base 288 with exponents > 60, it raises the question for me :

Do you think I should extend all bases < 1000 to exponent 70 on the project page ?
Please give me your opinions on this question, because the limit exponents I chose are not a rigorous choice.

For example for base 999, we have 999^70 which has 210 digits !
It's just to avoid nonsense.
I can't put 8128^100 on the page, because that number has 391 digits !

You don't need to expand the exponent.

[garambois]

Page updated.
Many thanks to all for your help !
Please let me know if you notice any errors.

Added bases : 14264, 14536, 15472 (C5 cycle completed !).
New bases reserved by yoyo : 39, 40, 44, 48, 50, 53, 62, 74, 162, 211, 722, 882.
Several terminated sequences of other bases with attribution.

I will be able to start doing the first data analysis tests of the project later this week...

Originally Posted by yoyo

You don't need to expand the exponent.

OK, Thank you for your answer.

Originally Posted by sweety439

OK, reserve base 120 (= 5!)

Thanks !
The reservations on the page will be effective only when the base will be initialized.

[RichD]

Looks like somebody might need more numbers to factor. I'll initialize bases 52, 54 & 55 next.

[richs]

392^57 terminates P42 at i27.

392^59 terminates P34 at i84.

[Happy5214]

Originally Posted by garambois

I understand that your criteria make you handle sequences that are not taken into account by our project page.
If you handle base 288 with exponents > 60, it raises the question for me :

Do you think I should extend all bases < 1000 to exponent 70 on the project page ?
Please give me your opinions on this question, because the limit exponents I chose are not a rigorous choice.

For example for base 999, we have 999^70 which has 210 digits !
It's just to avoid nonsense.
I can't put 8128^100 on the page, because that number has 391 digits !

IMO just extend everything beyond base 100 to a fixed digit start value (200 digits maybe?) and leave it at that, whatever exponent that is for that base. That seems round but rigorous enough, though we lose the roundness of the exponent limits themselves.

Originally Posted by garambois

Honestly, I am against this idea.
Indeed, if Catalan's conjecture is true, then the sequences of the main project will all come out of the main project one day.
The placement of a base in this hypothetical new category would therefore be in no way final, unlike the placements of bases in all our other current categories.
But I agree that creating a category for the Open-End bases of the main project would facilitate the statistics !
Because, in the other hand, it is very interesting indeed to calculate the sequences of the bases in the main project.
I had already noticed that for the Lehmer five, non-trivial sequences ending seemed to be very rare.
But I think this must be a coincidence.
Otherwise, we would have a very curious conjecture !
It's a great idea to dig into this question !

That answer seems contradictory. You first say the category is a bad idea, and then you say it could be a good idea? I'm confused. The set of open sequences is pretty steady, especially those below 1e5 (comparing the count on Wikipedia from mid-2015 to today's from the Blue Page, we've only eliminated 7 over that span, and none below 1e4), so while the set is not final, it is stable enough for long-term inclusion.

Edit: Unrelated, the credit on 882^44 needs to be changed to me. That was included in https://www.mersenneforum.org/showpo...postcount=1253.