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-   -   Aliquot sequences that start on the integer powers n^i (https://www.mersenneforum.org/showthread.php?t=23612)

 garambois 2021-08-09 18:21

OK, big scan started !
Bases still added : 648, 306, 396, 696, 780, 828, 888, 996 and also 120.
Many thanks to all.
Of these 9 additional bases, for the moment, I only plan to add the 648 base to the project page soon.
Please let me know when the initialization calculations for the other 8 bases are finished, so I can add them to the project page.

:smile:

 garambois 2021-08-09 18:28

[QUOTE=warachwe;585268]The problem is that when the factor 13 are with even power, it does not preserve the 7 for the second iteration.
This only happen when k is multiple of 13, for example 2^(12*13)-1 =3^2*5*7*13^2*53*79*...

This is why conjecture 34 ( 3^(18*37) ), 35 ( 3^(36*37) ), and 106 ( 11^(6*37) ) are false.

But this doesn't mean all similar conjecture are false, as there maybe others prime(s) that preserve p.

When we try to 'get rid of' those primes, there maybe yet another that will preserve p instead. Since the size of first iteration grow very quickly, it is hard to find other contradiction this way.

If some of those are true, I imagine the proof might be similar to the proof of conjecture (2).[/QUOTE]

Thank you so much for your explanations !
I think I'm going to have to add some red to the conjecture page very quickly !
I will do that in the next few days...

 Happy5214 2021-08-10 01:43

Conjecture [STRIKE]103[/STRIKE]104 is false (and doesn't even make any sense).

 garambois 2021-08-10 08:29

[QUOTE=Happy5214;585297]Conjecture [STRIKE]103[/STRIKE]104 is false (and doesn't even make any sense).[/QUOTE]
You are right.
Thank you very much for this comment.
It is an error in the statement.
I have corrected and the conjecture 104 becomes :
[I]The prime number 13 appears in the decomposition of the terms of index 1, 2 of all sequences that begin with the integers 11^(12*k).[/I]
I have verified it up to k=50.
But we will encounter with this conjecture the same problem as with the others of the same type.
Indeed, (11^12-1)/10 = 2^3*3^2*7*13*19*37*61*1117 and it is the number 1117 that maintains the 13 at the next iteration.
And, for the sequence 11^(12*1117), the factor 1117^2 appears in the factorization of the term in index 1.

[QUOTE=warachwe;585268]The problem is that when the factor 13 are with even power, it does not preserve the 7 for the second iteration.
This only happen when k is multiple of 13, for example 2^(12*13)-1 =3^2*5*7*13^2*53*79*...

This is why conjecture 34 ( 3^(18*37) ), 35 ( 3^(36*37) ), and 106 ( 11^(6*37) ) are false.

But this doesn't mean all similar conjecture are false, as there maybe others prime(s) that preserve p.

When we try to 'get rid of' those primes, there maybe yet another that will preserve p instead. Since the size of first iteration grow very quickly, it is hard to find other contradiction this way.

If some of those are true, I imagine the proof might be similar to the proof of conjecture (2).[/QUOTE]
Many thanks again for these observations !
- Conjecture (10) : Maintained, by luck ! Yes, the 13^2 factor appears if k is a multiple of 13, but by luck, there must be other primes that maintain the 7 factor at the second iteration.
- Conjecture (34) : invalidated.
- Conjecture (35) : invalidated.
- Conjecture (106) : invalidated.

 Happy5214 2021-08-10 13:36

I think the trick is to treat the 2 mod 3 in conjecture 2 not as an actual 2, but as -1 mod 3. That means you'd use 6 mod 7 for conjecture 10, giving you the prime 3121 as the one preserving the 13.

 sweety439 2021-08-13 11:28

[QUOTE=sweety439;584779]I looked [URL="http://factordb.com/sequences.php?se=1&aq=29%5E15&action=range&fr=0&to=60"]the Aliquot sequence of 29^15[/URL], this sequence is very interesting, although the first numbers are odd and these numbers decrease quickly, but the sequence reach an odd square number (265^2) and immediately merges with 18528.

I am curious of which number is the smallest odd number whose Aliquot sequence has not yet been fully determined? 1521 = 39^2 is the smallest odd number with long Aliquot sequence.

[COLOR="Red"]Edit: I have found it with my program, it is 3025 = 55^2[/COLOR][/QUOTE]

(although this may not appropriate for this thread, but I am quoting a post in this thread, and this is still related to Aliquot sequences)

I also found the smallest prime p such that the Aliquot sequence for 2*p has not yet been fully determined, this is p=2477 ([URL="http://factordb.com/sequences.php?se=1&aq=2*2477&action=range&fr=0&to=100"]Aliquot sequence for 2*2477 = 4954[/URL])

 garambois 2021-08-19 08:30

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

[B]Added bases : 51, 52, 54, 55, 276, 552, 564, 648, 660, 720, 966.[/B]
[B]New contributor to the calculation : henryzz as HRZ, that we already knew for his contribution in theory ![/B]
[B]Several finished sequences have been added on old bases, sometimes quite small (5, 6, 7...)[/B]

I have not yet added the bases 120, 306, 396, 696, 780, 828, 888 and 996.
Let me know if someone finishes initializing these 8 new bases.

Unless I'm mistaken, for all bases smaller than 9699690, there are only bases 276, 552, and 966 (all three are Lehmer five sequences) that do not have non-trivial sequence ends.
But this must be a coincidence, because we have small bases like base 41 that have only one non-trivial sequence ending.
We will still have to observe this closely.
And we will have to calculate the odd exponent sequences of bases 276, 552 and 966 to find some that end on a cycle or a prime number !

 garambois 2021-08-19 09:30

1 Attachment(s)

It is now 10 days since I launched the big scan of the project to analyze the data.
And I've been looking at the data from all angles for days.
I was even seeing numbers at night in my sleep !
And then : NOTHING !
While analyzing the data last year and again a few months ago, I had observed some remarkable phenomena that led to the 140 conjectures you know.
But these easily remarkable and "obvious" things had not been foreseen and had been observed by chance while I was looking for something else.

[U]What exactly am I looking for[/U] ?

This project was originally created to try to see if a sequence which starts with a number which is an integer power of a number was more likely to belong to such or such "branch" of the [URL="http://www.aliquotes.com/graphinfinisuali.htm"]infinite graph of aliquot sequences[/URL].
For a given base [I]b[/I] and an integer [I]a[/I] to find, I was looking to observe things like :
"Sequences of the type [I]b[/I] ^ ([I]k[/I] * [I]a[/I]) end with the prime number [I]p[/I] (or with the cycle [I]c[/I]) for any integer [I]k[/I]. "
I am well aware that it is highly unlikely that such a conjecture could be formulated, but I have not finished looking.
Perhaps there is a rule to be found which is more complicated to formulate than this example.
But for now, I can't see where to look, I have to take a break.
But maybe I also missed something obvious ?

So, in attached files, I put at your disposal my observation tables : perhaps you will see something there that escaped me ?
Perhaps you will also find a way to visualize the data differently than I do to reveal interesting things ?
Good luck to you in your observations of these 4 big tables.

When it comes to cycles that end sequences, we still don't have enough to be able to do statistics.
I manually noted the data on paper with a pencil to make my observations : NOTHING EITHER !

Later, I will again make observations in other directions, outside of the original idea of the project, to try to find something else entirely.
But I keep in mind that the holy grail of this project would be to succeed in predicting the end of an aliquot sequence without having to calculate all the terms.
I'm just assuming that this is easier to do for sequences that start with whole powers than for all sequences in general.

Do not hesitate if you have any comments, or even criticisms to make following this post.
Maybe I need someone to bring me to my senses...

[B][SIZE=3];-)[/SIZE][/B]

 yoyo 2021-08-19 17:14

I'll take bases 276, 552 and 966.

 bur 2021-08-20 07:16

Whatever may come from it, thanks for your post because it clarified the original goal in a concise way even I understood.

My conclusion from very limited mathematical background is, if for so many bases no regularities regarding the termination were observed, then probably there are none? Or rather, they are so complicated and comprise so many cases, it's not possible to see from a limited sample size.

I know math is not an empiric science, so "probably" doesn't count as much as in other fields, but still.

As a general point of view, if a scientific project seems to go to nothing, then take a break from it. If after a few weeks or months you feel like there is still something to it, pick it up again. But don't drive yourself crazy with it.

And as a last remark, the project still served and serves a purposes. For some it's mainly an interesting pasttime, some might see it in light of compiling a list, of completing things, then some conjectures seem to have come from it and so on. It's certainly not worse than spending months of core-time on factoring a single huge number just to add one more data point to something. So please keep it running even if you take a break from analyzing. No one analyzes the sequences from the Blue Page. The database you created is a great thing.

 VBCurtis 2021-08-20 17:03

[QUOTE=bur;586110]Whatever may come from it, thanks for your post because it clarified the original goal in a concise way even I understood.

And as a last remark, the project still served and serves a purposes. For some it's mainly an interesting pasttime, some might see it in light of compiling a list, of completing things, then some conjectures seem to have come from it and so on. It's certainly not worse than spending months of core-time on factoring a single huge number just to add one more data point to something. So please keep it running even if you take a break from analyzing. No one analyzes the sequences from the Blue Page. The database you created is a great thing.[/QUOTE]

I 100% agree with this! :tu:

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