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

 garambois 2020-08-23 19:02

Thank you for extending the calculations for Base 2. I'll make the change at the next update. It's a fundamental base.

It's a great idea to create all the lists of prime numbers at the end of the sequences of all the bases, like for base 79 ! I'm working on that too.
The main problem is to display everything as legibly as possible, so that we can see things. I still don't know which elements are important !
Maybe you need to show the prime exponent factorizations for each base or something like that ? It's up to us to be creative...
Please, if it's possible, for example for base 79, display it more like this, so that we can see more information at only a glance :

[CODE]7 (1)
11 (1)
13 (1)
19 (1)
23 (1)
...
41 (3)
...[/CODE] should be :
[CODE]7 (i1=p1^a1*p2^a2*...)
11 (i1=p1^a1*p2^a2*...)
13 (i1=p1^a1*p2^a2*...)
19 (i1=p1^a1*p2^a2*...)
23 (i1=p1^a1*p2^a2*...)
...
41 (i1=p1^a1*p2^a2*...)
41 (i2=p1^a1*p2^a2*...)
41 (i3=p1^a1*p2^a2*...)
...[/CODE]where i1, i2, i3 are, for example, the three exponents of the sequences ending with the prime number 41 and for each of these three exponents its factorization in prime numbers appears.

 EdH 2020-08-23 20:40

1 Attachment(s)
[QUOTE=garambois;554744]Thank you for extending the calculations for Base 2. I'll make the change at the next update. It's a fundamental base.

It's a great idea to create all the lists of prime numbers at the end of the sequences of all the bases, like for base 79 ! I'm working on that too.
The main problem is to display everything as legibly as possible, so that we can see things. I still don't know which elements are important !
Maybe you need to show the prime exponent factorizations for each base or something like that ? It's up to us to be creative...
Please, if it's possible, for example for base 79, display it more like this, so that we can see more information at only a glance :

[CODE]7 (1)
11 (1)
13 (1)
19 (1)
23 (1)
...
41 (3)
...[/CODE] should be :
[CODE]7 (i1=p1^a1*p2^a2*...)
11 (i1=p1^a1*p2^a2*...)
13 (i1=p1^a1*p2^a2*...)
19 (i1=p1^a1*p2^a2*...)
23 (i1=p1^a1*p2^a2*...)
...
41 (i1=p1^a1*p2^a2*...)
41 (i2=p1^a1*p2^a2*...)
41 (i3=p1^a1*p2^a2*...)
...[/CODE]where i1, i2, i3 are, for example, the three exponents of the sequences ending with the prime number 41 and for each of these three exponents its factorization in prime numbers appears.[/QUOTE]
I'm not sure I understand this fully at first glance, but I'll work on it.:smile:

For now, attached is a listing for all primes <1000 across all the tables, showing which sequences they terminate, if any. For example, here's a sample:
[code]
Prime 2:
============
2^1

Prime 3:
============
2^164
2^2
2^305
2^317
2^4
2^55
3^1
3^247
3^2
3^5
5^38
6^152
7^4
7^77
11^15
11^2
12^1
12^2
13^15
14^76
14^80
15^1
19^15
21^21
21^55
23^3
30^1

Prime 5:
============
5^1

Prime 7:
============
2^10
2^12
2^141
. . .
Prime 41:
============
2^112
2^117
2^23
2^281
2^373
2^405
2^411
2^47
2^6
2^8
6^13
6^5
7^11
11^14
11^57
12^32
12^56
15^2
15^6
17^65
18^39
24^18
79^17
79^2
79^4
210^4
385^11
439^2
770^13
2310^8
. . .
Prime 149:
============
3^55

Prime 151:
============
18^79
210^40

Prime 157:
============

Prime 163:
============
3^47

Prime 167:
============

Prime 173:
============
7^31
. . .
[/code]Sorry, they are only sorted by base, not power.

 EdH 2020-08-24 00:45

1 Attachment(s)
@Jean-Luc:

Would this be more toward what you need:
[code]
Listing of all terminating primes <1000 across all tables

Prime 2:
============
2^(1)

Prime 3:
============
2^(2^2*41)
2^(2)
2^(5*61)
2^(317)
2^(2^2)
2^(5*11)
3^(1)
3^(13*19)
3^(2)
3^(5)
5^(2*19)
6^(2^3*19)
7^(2^2)
7^(7*11)
11^(3*5)
11^(2)
12^(1)
12^(2)
13^(3*5)
14^(2^2*19)
14^(2^4*5)
15^(1)
19^(3*5)
21^(3*7)
21^(5*11)
23^(3)
30^(1)

Prime 5:
============
5^(1)

Prime 7:
============
2^(2*5)
2^(2^2*3)
2^(3*47)
2^(2*139)
. . .

[/code]I've attached a new copy of the last file of primes <1000 across all tables.

 yoyo 2020-08-24 05:34

[QUOTE=EdH;554704]I have extended base 2 up through [I]i[/I]=549. I can work a few more there (not sure if I'll make it to 559), but if yoyo wants to take over with base 2, that would be fine.
[/QUOTE]

I would take base 2 from \$i=540 to \$i <= 559 until the remaining composite is > C139.

yoyo

 garambois 2020-08-24 07:22

@yoyo : Many thanks !

@EdH : Is it possible to have :
[CODE]Prime 3: ============
2^164 = 2^(2^2*41)
2^2 = 2^(2)
...
[/CODE][U]But sorted[/U] !
No problem if it's too hard to do, I'll manage otherwise!

 EdH 2020-08-24 11:48

[QUOTE=garambois;554790]@yoyo : Many thanks !

@EdH : Is it possible to have :
[CODE]Prime 3: ============
2^164 = 2^(2^2*41)
2^2 = 2^(2)
...
[/CODE][U]But sorted[/U] !
No problem if it's too hard to do, I'll manage otherwise![/QUOTE]
I was able to sort the bases, but the powers proved more difficult because the normal sort works by character, which means 164 is before 2, because 1 is before 2. I'll work on the formatting, at least.

 RichD 2020-08-24 12:59

[CODE]Prime 3: ============
2 ^ 164 = 2^(2^2*41)
2 ^ 2 = 2^(2)
...[/CODE]
Maybe add spaces and numeric sort the third field. Not pretty.

 EdH 2020-08-24 13:08

1 Attachment(s)
Here is a listing with the new formatting. I still haven't achieved the secondary (powers) sorting. These listings currently are only based on all primes below 1000. They include the mention of primes which have not been found at all as terminations. I'm running a current process to update and list only primes that are found as terminations across all tables. It's taking time to check for currency of all the sequences to see if any have terminated since the last check. Once I have the full list of all primes that terminate sequences, I will run that list for a total document of all terminating primes across all the tables on the page. That may still be a couple days.

 yoyo 2020-08-24 17:39

I take also base 10 and 11 until a composite is > C139.

 garambois 2020-08-24 17:48

Okay, thank you very much Ed.
This is exactly the document we need !
I'm also wondering why from the prime number 337, we can't find the factorizations of the exponent anymore ?

For my part, I'm currently trying to understand why we don't have an Open-End sequence for bases 2, 18 and the other bases for the exponents that have the same parity as them...

 EdH 2020-08-24 19:47

1 Attachment(s)
Here's a full list of all the terminating primes from all the sequences across all the tables on the page. I also added base 79 through power 79 since it's ready to add to the page. There are 1692 unique terminating primes found to this date.

I still haven't solved the exponent sorting, but the bases should be sorted.

 EdH 2020-08-25 01:39

[QUOTE=garambois;554843]. . .
I'm also wondering why from the prime number 337, we can't find the factorizations of the exponent anymore ?
. . .
[/QUOTE]Although they are there for further, they are still missing after a point. I am trying to find out why, but not getting anywhere, yet. Oddly they stop during the 601 prime, at the line right after the 601 line of the document.:confused:

 EdH 2020-08-25 02:04

1 Attachment(s)
[QUOTE=EdH;554871]Although they are there for further, they are still missing after a point. I am trying to find out why, but not getting anywhere, yet. Oddly they stop during the 601 prime, at the line right after the 601 line of the document.:confused:[/QUOTE]
Alright! I found the culprit. I think this document will be complete. The 601 was a subtle indicator - an easy fix after all, once I realized the real problem.

 RichD 2020-08-25 02:32

Table n=20 will be ready to insert about the time the next update is due to be posted.

 garambois 2020-08-25 09:39

@EdH : Many thanks to you. The table is perfect. All I have to do now is to do by hand, the last sort in ascending order of exponents, while waiting to write my own program that will give the same output.
But for the moment at the sight of this table, it does not seem to appear of new conjecture !
Unfortunately, the start of the school year is fast approaching and I would have less time to work on on the aliquot sequences !

@RichD : Thanks a lot. I don't think I'll do an update until next weekend. I will then add bases 79 and 20...

 garambois 2020-08-25 10:50

[SIZE=4]About the search for possible Open-End sequences among those that should end trivially[/SIZE]

The sequences we have calculated so far that end trivially are those of bases 2, 18 and those beginning with b^i with b (base) and i (exponent) of the same parity. For all these sequences, at the first iteration, we get an odd number. This is what makes them end. All the following terms remain essentially odd, with rare exceptions where we always end up with an odd number. For one of these aliquot sequence to be Open-End, one of its odd terms would have to be an aliquot antecedent of an even Open-End aliquot sequence, which we are familiar with in the main project.

Let me give you the following two facts, and after, draw the conclusions :

1) At the last update on August 21, 2020, in our project, we had calculated 1827 sequences that ended trivially.

2) Using my [URL="http://www.aliquotes.com/aliquote_base.htm#alibasefonda"]fundamental database[/URL], I determined the following :
- There are 4 odd numbers less than 10,000 which are the start of Open-End sequences (3025, 7225, 8015 and 8427)
- There are 80 odd numbers less than 100,000 which are the start of Open-End sequences.
- There are 810 odd numbers less than 1,000,000 which are the start of Open-End sequences.
- There are 7734 odd numbers less than 10,000,000 which are the start of Open-End sequences.
In fact, these numbers are a bit too big, because in my fundamental database, a sequence is Open-End as soon as its terms reach 50 digits.
But the order of magnitude is there: out of all the odd numbers, there is 1 in 1300 or even say 1 in 2000, which is the start of an Open-End sequence.

Taking into account these two facts, and as in a sequence, one "picks" an odd number at each iteration, I ask the question that causes me so much trouble :

[B]Why hasn't an Open-End sequence been found yet among those that must end trivially ?[/B]

 garambois 2020-08-25 12:30

Please, on the previous post, I spotted an error and I can no longer correct myself !
[CODE]- There are 810 odd numbers less than 100,000 which are the start of Open-End sequences.
- There are 7734 odd numbers less than 1,000,000 which are the start of Open-End sequences.
In fact, these numbers are a bit too big, because in my fundamental database, a sequence is Open-End as soon as its terms reach 50 digits.
But the order of magnitude is there: out of all the odd numbers, there is 1 in 130 or even say 1 in 200, which is the start of an Open-End sequence.[/CODE]Should be :
[CODE]- There are 810 odd numbers less than 1,000,000 which are the start of Open-End sequences.
- There are 7734 odd numbers less than 10,000,000 which are the start of Open-End sequences.
In fact, these numbers are a bit too big, because in my fundamental database, a sequence is Open-End as soon as its terms reach 50 digits.
But the order of magnitude is there: out of all the odd numbers, there is 1 in 1300 or even say 1 in 2000, which is the start of an Open-End sequence.[/CODE]Many thanks !

 garambois 2020-08-25 17:30

Thank you so much Ed !

 yoyo 2020-08-25 19:41

I take base 12.
Do I take too much?
On base 13 are some reservations.

 VBCurtis 2020-08-25 19:51

I've been working on base 13 for a few years, and prefer to work only on base 13. Please skip that one!

 EdH 2020-08-25 20:30

1 Attachment(s)
OK! I think I have them all sorted correctly, now. Attached is a new document with bases AND powers sorted (and, the power expansion is to the end).

 garambois 2020-08-26 08:20

[QUOTE=yoyo;554932]I take base 12.
Do I take too much?
On base 13 are some reservations.[/QUOTE]

OK, thanks a lot !
At the next update, I will reserve the base 12 for you.
No, you don't take too much !
It all depends on YAFU's computing means.
YAFU's statistics are very high at the moment.
I don't know if a team is raiding on YAFU and if the statistics will stay so high in the next few times ?

 garambois 2020-08-26 08:48

[QUOTE=EdH;554939]OK! I think I have them all sorted correctly, now. Attached is a new document with bases AND powers sorted (and, the power expansion is to the end).[/QUOTE]

Ok, this time the file is perfect !
Thanks a lot Ed.

I examined it closely.
Unfortunately, nothing special catches my attention at the moment.
Except a detail for the prime number 53 : there are only sequences that start on powers of 2 that end with the prime number 53. But this must be pure chance !

But maybe someone else will observe something interesting...

:smile:

 garambois 2020-08-26 09:46

I would also like to make a comment about a private conversation that Edwin Hall and I had in early July. Edwin allowed me to talk about this private conversation here when the messages are readable by everyone.
Here is Edwin's observation :

[CODE]Sequences that had abundant indices somewhere and a parity change other than at index 1 (all were due to perfect squares):
(2^9, 2^62, 2^210): 81 >> 40
(2^12, 2^141, 2^278, 2^387): 49 >> 8
(2^112): 2209 >> 48
(2^117): 25921 >> 5600
(2^141): (729 >> 364) and (49 >> 8)
(2^243): 1225 >> 542
(2^271): 2025 >> 1726
(2^305, 2^317): 9 >> 4
(2^421): 169 >> 14
Of the even numbers that were due to parity flips, only 40, 48, 364 and 5600, are abundant.[/CODE]This means, for example, that sequences that begin with 2^9, 2^62 and 2^210 arrive on the integer 81, which is a perfect square and therefore, at the next iteration, we will have an even term (here, 40, because s(81)=40). The goal was to find parts of sequences that begin with powers of 2 and that are increasing. The terms are then abundant.

I think we can generalize this study by looking at the table called "base_2_mat" that I attached to post #447.
Here is an excerpt from this table :
[CODE]prime 2 in sequence 2^1 at index 0
prime 2 in sequence 2^2 at index 0
prime 2 in sequence 2^3 at index 0
prime 2 in sequence 2^4 at index 0 3
...
prime 2 in sequence 2^9 at index 0 3 4
prime 2 in sequence 2^10 at index 0 5
...
prime 2 in sequence 2^12 at index 0 8
...
prime 2 in sequence 2^14 at index 0 7 8
prime 2 in sequence 2^15 at index 0 7 8
...
prime 2 in sequence 2^55 at index 0 14
...
prime 2 in sequence 2^59 at index 0 12 13
...
prime 2 in sequence 2^62 at index 0 26 27
...
prime 2 in sequence 2^112 at index 0 62 63 64
...
prime 2 in sequence 2^117 at index 0 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56
...
prime 2 in sequence 2^141 at index 0 25 26 27 28 34
...
prime 2 in sequence 2^164 at index 0 42
...
prime 2 in sequence 2^210 at index 0 44 45 46 49 50
...
prime 2 in sequence 2^243 at index 0 128 129 130 131
...
prime 2 in sequence 2^271 at index 0 79 80 81 82 83 84 85 86 87 88 89
...
prime 2 in sequence 2^278 at index 0 51
...
prime 2 in sequence 2^305 at index 0 76
...
prime 2 in sequence 2^317 at index 0 70
...
prime 2 in sequence 2^373 at index 0 94 95 96 97
...
prime 2 in sequence 2^387 at index 0 102
...
prime 2 in sequence 2^421 at index 0 65 66 67
...
prime 2 in sequence 2^510 at index 0 125 126
...[/CODE]For more readability, I removed all the lines where there was only the index 0 of concerned. There are dotted lines instead. When you see this table, you can immediately see in which sequences from base 2, the prime number 2 appears in the factorization of terms at an index other than 0. All sequences with increasing parts identified by Edwin are found in this way, plus others: those with even terms but which are in "downdriver".

[B]Unfortunately, here again: I don't notice anything about the exponents, nor the indexes that could allow to predict for which base 2 exponents, we have parts of sequences with the prime number 2 elsewhere than at the index 0 !!![/B]

 RichD 2020-08-26 13:05

I'll start work on Table n=29.

 RichD 2020-08-26 13:37

Tabe n=29

This may be of interest because it flips parity.

[url]http://factordb.com/sequences.php?se=1&aq=29%5E15&action=range&fr=32&to=42[/url]

 Sergiosi 2020-08-26 13:37

Reserving 2310^25

 garambois 2020-08-26 15:47

@RichD :
OK for base 29.
OK for 265^2, thank you very much ! This number was already in my tables...

@Sergiosi :
I think you have already finished the calculations for 2310^25. And moreover, it is a non-trivial end, which is rare ! Thanks a lot !

 RichD 2020-08-27 20:40

Who has the way back merge detection meter? I believe 20^37 has merged.

 EdH 2020-08-27 21:27

[QUOTE=RichD;555154]Who has the way back merge detection meter? I believe 20^37 has merged.[/QUOTE]
20^37:i1855 merges with 660:i25

 richs 2020-08-27 22:37

[QUOTE=richs;553180]Reserving 439^30 at i80.[/QUOTE]

439^30 is now at i124 (added 44 iterations) and a C121 level with a 2^6 * 3 guide, so I will drop this reservation. The remaining C119 term is well ecm'ed and is ready for nfs.

Taking 439^36 at i68.

 RichD 2020-08-28 01:14

19^14 might have also merged.

 EdH 2020-08-28 02:35

[QUOTE=RichD;555203]19^14 might have also merged.[/QUOTE]
True![code]19^14:i765 merges with 755460:i25[/code]

 EdH 2020-08-28 15:05

@RichD:
Not sure if these have already been mentioned*, but there are a couple other base 20 merges:
[code]
20^7:i60 merges with 7044:i129
20^9:i27 merges with 709900:i10
[/code]
[SIZE=2]*Sorry, if this is "old news."[/SIZE]

 RichD 2020-08-28 16:50

I remembered a low exponent but couldn't remember the base. Great that you found them.

Also, 23^44 appears to merge.

 EdH 2020-08-28 18:28

[QUOTE=RichD;555289]I remembered a low exponent but couldn't remember the base. Great that you found them.

Also, 23^44 appears to merge.[/QUOTE]
It does:
[code]
23^44:i1830 merges with 7560:i251[/code]

 RichD 2020-08-29 10:52

Table n=29 is ready to insert on the webpage when the update comes around. I took it to i=83.

 EdH 2020-08-29 13:33

[QUOTE=RichD;555366]Table n=29 is ready to insert on the webpage when the update comes around. I took it to i=83.[/QUOTE]
In case they've not been noted earlier:
[code]
29^6:i51 merges with 2441868:i4
29^15:i38 merges with 18528:i1
[/code]

 EdH 2020-08-30 12:26

In case it's of use, here is a compilation of the merges not yet shown in the tables:
[code]
19^14:i765 merges with 755460:i25

20^7:i60 merges with 7044:i129
20^9:i27 merges with 709900:i10
20^37:i1855 merges with 660:i25

23^12:i144 merges with 78660:i2
23^44:i1830 merges with 7560:i251

770^3:i29 merges with 165876:i98
[/code]

 garambois 2020-08-30 18:56

OK, page updated.
A lot of thanks to all for your help.
The amount of calculations performed in 10 days was colossal !

I thank you to check if everything is correct for the work concerning you...

Bases 20, 29 et 79 added.
And more details...

:smile:

 EdH 2020-08-30 22:23

@Jean-Luc:
Now that we have a clearer picture of all the added tables, is there something in particular that you would prefer I work on?

If not, I'll go ahead and dabble in the base 79 table for a bit - no reservations necessary, for now.

Thank you for all the work keeping the tables straight!

 RichD 2020-08-31 13:14

The tables are really nice and helpful.

The following sequences terminate.
20^82
29^84
20^86
20^88
20^90
23^10
23^77
23^79
23^81
29^63

 garambois 2020-08-31 16:02

[QUOTE=RichD;555523]The tables are really nice and helpful.

The following sequences terminate.
20^82
29^84
20^86
20^88
20^90
23^10
23^77
23^79
23^81
29^63[/QUOTE]

Okay, thank you very much.
This will be taken into account in the next update probably next weekend.

 garambois 2020-08-31 17:00

[QUOTE=EdH;555489]@Jean-Luc:
Now that we have a clearer picture of all the added tables, is there something in particular that you would prefer I work on?

If not, I'll go ahead and dabble in the base 79 table for a bit - no reservations necessary, for now.

Thank you for all the work keeping the tables straight![/QUOTE]

This time, I had about 3 - 4 hours of work to do the update !
:smile:
It is likely that at the next update, I won't touch any more the bases where nobody reserved sequences. I will update these bases only once every 2 or 3 months.

I think that the best thing for the moment is indeed to continue the calculations of the different bases that we have started. We will see how much new data we will have in a few weeks, months, or even a year.
It all depends on how many volunteers are going to do the calculations.

But everyone should be well aware that we may not find what we were looking for at the beginning. This project was born in the following way. When with friends we were looking at the infinite graph of the aliquot sequences ([URL="http://www.aliquotes.com/graphinfinisuali.htm"]see the graph here[/URL]), we wondered if among the sequences starting with integer powers of a prime number, there would be more that would end with the same prime number. For example, would sequences starting with 7^i (i=1, 2, 3...) end more often with the prime number 7 ?
Today, we cannot answer this question.
But we have found something else that we were not looking for at all at the beginning : the 133 conjectures of post #447. One never knows to which ideas the chance of reflections can lead us ?
Maybe all the calculations we make will allow us to answer the original question. Maybe not ! Or maybe we'll again find something else that we didn't expect to find.
But to find things, we have to look at the data and ask ourselves all sorts of questions. Anyone can do that... You have to be as creative as possible.

But right now, I'm still waiting for an Open-End sequence in base 2, 18 or another base with an exponent that has the same parity as the base.

 EdH 2020-08-31 19:18

@Jean-Luc:
Thanks for all the work and explanation. I will possibly play with extending base 18, if RichD isn't working there anymore and then play randomly with the matched parity unreserved sequences in the various tables.

But, as we study these, I'm beginning to get a different feeling for "open" sequences. Open really only means we haven't finished it. 2^552, after index 0, had 9 abundant indices. If, when it reached 168 digits, we stopped, it would have been considered "open" and would have met the search criteria.

@RichD:
Are you still doing work on base 18?

 RichD 2020-08-31 22:02

[QUOTE=EdH;555579] @RichD:
Are you still doing work on base 18?[/QUOTE]

All done with n=18. It's all yours.

I am slowing down on the others but still randomly doing a bit of work on n=19, 20, 23, 29, 770.

 VBCurtis 2020-08-31 22:11

[QUOTE=EdH;555579] But, as we study these, I'm beginning to get a different feeling for "open" sequences. Open really only means we haven't finished it.[/QUOTE]
This is the original hypothesis of the Aliquot-sequence project- that they all terminate, eventually. We keep cracking C190+ cofactors in hopes of breaking a driver and finding a new largest-peak-size-that-still-terminates record.

 EdH 2020-09-01 00:52

[QUOTE=RichD;555588]All done with n=18. It's all yours.
. . .[/QUOTE]Thanks - I'll work there for a while, then.

[QUOTE=VBCurtis;555590]This is the original hypothesis of the Aliquot-sequence project- that they all terminate, eventually. We keep cracking C190+ cofactors in hopes of breaking a driver and finding a new largest-peak-size-that-still-terminates record.[/QUOTE]When I first started playing with Aliquot sequences, I was in the Guy-Selfridge camp. But, since I've been working on this sub-project, I'm swinging much more toward the Catalin-Dickson Conjecture. However, I have to keep in mind that we are really working on a tiny piece of the "big" picture by working with bases and powers. Then again, if all bases and powers terminate, then everything does, because all numbers can be bases.

 garambois 2020-09-01 16:44

[QUOTE=EdH;555579]
But, as we study these, I'm beginning to get a different feeling for "open" sequences. Open really only means we haven't finished it. 2^552, after index 0, had 9 abundant indices. If, when it reached 168 digits, we stopped, it would have been considered "open" and would have met the search criteria.
[/QUOTE]

For 2^552, yes : 9 abundant indices.
But not 9 even terms !

Just for information : I'm in the Guy-Selfridge camp and I think that one day we'll be able to build a number that we'll be sure is the start of an Open-End sequence that will grow endlessly...

 EdH 2020-09-02 00:56

[QUOTE=garambois;555669]For 2^552, yes : 9 abundant indices.
But not 9 even terms !

Just for information : I'm in the Guy-Selfridge camp and I think that one day we'll be able to build a number that we'll be sure is the start of an Open-End sequence that will grow endlessly...[/QUOTE]
Ah, so we need even terms for it to be open, as well, because odd terms will eventually descend?

I won't hold that against you.:smile: I might even move back that way myself. But, currently, we seem to be displaying a lot of evidence for terminations.

 EdH 2020-09-02 00:59

[QUOTE=VBCurtis;555590]This is the original hypothesis of the Aliquot-sequence project- that they all terminate, eventually. We keep cracking C190+ cofactors in hopes of breaking a driver and finding a new largest-peak-size-that-still-terminates record.[/QUOTE]But, we need some new algorithms. My attention span is short. I can't reasonably factor a difficult c190.

 VBCurtis 2020-09-02 03:16

[QUOTE=EdH;555725]I can't reasonably factor a difficult c190.[/QUOTE]

That's what nfs@home is for. We merely need to ECM-pretest and solve the matrix, a total of at most 20% of the sieve time.

Edit: Of course, all that does is delay the onset of not-enough-attention-span to ~C200.

 garambois 2020-09-02 19:23

[QUOTE=EdH;555724]Ah, so we need even terms for it to be open, as well, because odd terms will eventually descend?

I won't hold that against you.:smile: I might even move back that way myself. But, currently, we seem to be displaying a lot of evidence for terminations.[/QUOTE]

In fact, I don't really know what we should consider an Open-End sequence. But I do make two observations :

1) In the main project, on the blue page, I've never seen an Open-End sequence that starts on an odd number. But maybe there is one and I don't know how to filter it ?

2) On all the aliquots sequences that start on integers from 1 to 10,000,000, thanks to my [URL="http://www.aliquotes.com/aliquote_base.htm#alibasefonda"]fundamental database[/URL], I know that the absolute record for a growth sequence with consecutive odd terms is held by the sequence that starts on the integer 2,551,185. And this number of growing consecutive terms is only 6, which is very little. ([URL="http://www.aliquotes.com/aliquote_base_fond_applic.htm"]see here some other records[/URL])

Having made these two observations, I tell myself that a sequence can only be Open-End with even terms for numbers of a few hundred digits that we work with.
But maybe it should also be possible to have an Open-End sequence with huge odd terms. But discovering such odd Open-End sequences (if they exist) is far from being within our reach.

 EdH 2020-09-06 17:57

In case you do an update this weekend prior to my finishing 18^119, I have otherwise completed a new row for base 18, through 18^118 at present.

I am running 18^119 and hope to have it done soon. For now, I will not be extending the table further.

 richs 2020-09-06 19:01

[QUOTE=richs;555182]Taking 439^36 at i68.[/QUOTE]

439^36 is now at i148 (added 80 iterations) and a C130 level with a 2^6 * 3 * 5 * 31 guide, so I will drop this reservation. The remaining C126 term is well ecm'ed and is ready for nfs.

Taking 439^38 at i52.

 garambois 2020-09-06 19:27

OK, page updated.
A lot of thanks to all for your help.

:smile:

[QUOTE=EdH;556250]In case you do an update this weekend prior to my finishing 18^119, I have otherwise completed a new row for base 18, through 18^118 at present.

I am running 18^119 and hope to have it done soon. For now, I will not be extending the table further.[/QUOTE]

Sorry Edwin, but I had done the update before reading your message! But next week, I'm going to extend base 18 to 18^129.
do you want to make reservations for this base ?

In the same way, I think to add the bases 50 (=2*5^2), 72 (=2*6^2) and 98 (=2*7^2), up to 50^95, 72^88 and 98^82.
It is an effective way to have many sequences that end because for these three bases, as for bases 2 and 18, all sequences end.

I take care of base 50 and I will add it only when I will have finished all the sequences up to 50^72.
If someone wants to do the preliminary work for bases 72 and 96, let me know here.
I believe that with these three additional bases, we will be very well equipped for our statistics in a few months...

[QUOTE=RichD;555588]
I am slowing down on the others but still randomly doing a bit of work on n=19, 20, 23, 29, 770.[/QUOTE]

RichD, do you want to make reservations for these bases ?

[QUOTE=richs;556259]439^36 is now at i148 (added 80 iterations) and a C130 level with a 2^6 * 3 * 5 * 31 guide, so I will drop this reservation. The remaining C126 term is well ecm'ed and is ready for nfs.

Taking 439^38 at i52.[/QUOTE]

OK, got it. Thank you very much.
I'll change that in the next update.

 garambois 2020-09-06 19:35

Please, do you manage to reproduce the same error as me, when you enter on factordb the sequence 72^89 from index 0 to 1, like this :

[URL]http://factordb.com/sequences.php?se=1&aq=72%5E89&action=range&fr=0&to=1[/URL]

The answer is totally absurd, at index 0, it indicates 2^534 !!!

 EdH 2020-09-06 21:32

[QUOTE=garambois;556265]Please, do you manage to reproduce the same error as me, when you enter on factordb the sequence 72^89 from index 0 to 1, like this :

[URL]http://factordb.com/sequences.php?se=1&aq=72%5E89&action=range&fr=0&to=1[/URL]

The answer is totally absurd, at index 0, it indicates 2^534 !!![/QUOTE]
There are several others, as well:
[code]
72^71 shows 2^426
72^73 shows 2^438
72^79 shows 2^474
72^83 shows 2^498
72^79 shows 2^474
[/code]I didn't check the whole range.

I was going to offer to do the preliminaries for 72. Perhaps I should choose 98.:smile:

I had not intended to extend base 18 past 2^119 (at least, not for now), so others are free to reserve those.

 RichD 2020-09-06 21:51

[QUOTE=garambois;556263] RichD, do you want to make reservations for these bases ?[/QUOTE]

No, just 770. I'll take the remainder to orange, if that is where they go...

 RichD 2020-09-07 09:06

It looks like Table n=23 didn't get updated which includes these from above.
[QUOTE=RichD;555523]The following sequences terminate.
. . .
23^10
23^77
23^79
23^81[/QUOTE]
19^91
20^92

 garambois 2020-09-07 16:57

[QUOTE=EdH;556277]There are several others, as well:
[code]
72^71 shows 2^426
72^73 shows 2^438
72^79 shows 2^474
72^83 shows 2^498
72^79 shows 2^474
[/code]I didn't check the whole range.

I was going to offer to do the preliminaries for 72. Perhaps I should choose 98.:smile:

I had not intended to extend base 18 past 2^119 (at least, not for now), so others are free to reserve those.[/QUOTE]

OK, thank you for checking for the malfunction for base 72. Under these conditions, it is better to leave this base aside for the moment !
A few months ago, I had reported other malfunctions on factordb, on sequences of the main project ([URL="https://mersenneforum.org/showthread.php?p=544399"]see here[/URL]). But the errors were never corrected, nor reported on the blue page. So this is likely to last ! It is indeed better to calculate the base 98 !

OK, seen for base 18.

 garambois 2020-09-07 16:59

[QUOTE=RichD;556278]No, just 770. I'll take the remainder to orange, if that is where they go...[/QUOTE]

OK, I'll reserve the 770 base for you at the next update. Thank you very much.

 garambois 2020-09-07 17:02

[QUOTE=RichD;556312]It looks like Table n=23 didn't get updated which includes these from above.

19^91
20^92[/QUOTE]

I must have made a mistake on base 23. I think you're right, it hasn't been updated.
I'm sorry !
I will correct it in the next update.

 EdH 2020-09-07 18:38

[QUOTE=garambois;556348]OK, thank you for checking for the malfunction for base 72. Under these conditions, it is better to leave this base aside for the moment !
A few months ago, I had reported other malfunctions on factordb, on sequences of the main project ([URL="https://mersenneforum.org/showthread.php?p=544399"]see here[/URL]). But the errors were never corrected, nor reported on the blue page. So this is likely to last ! It is indeed better to calculate the base 98 !

OK, seen for base 18.[/QUOTE]
I have base 98 underway, but I'll probably only take it up to index 0 >= c140 dd.

I intend to work with base 72 also (to the same size as base 98), with the intention of providing the correct .elfs as attachments for those that are incorrect in the db. That way, we can still do data harvesting via local files. I have posted the list of incorrect ones (checked up through 72^100) [URL="https://www.mersenneforum.org/showpost.php?p=556327&postcount=447"]here[/URL]. For the >=140 sequences that are incorrect in the db, I'll supply .elfs which contain at least indices 0 and 1, so that detail of study can be continued.

At a later time, I'll revisit the higher exponents for the tables for bases 18, 72, 98. Maybe someone else may be interested in the higher ends of those tables by then.

 garambois 2020-09-10 17:05

[QUOTE=EdH;555372]In case they've not been noted earlier:
[code]
29^15:i38 merges with 18528:i1
[/code][/QUOTE]

After a few checks on the data, I would note that :
[code]
29^15:i38 merges with 18528:i1
[/code]should be :
[code]
29^15:i37 merges with 18528:i0
[/code]Do you confirm that I should make this change ?

 EdH 2020-09-10 17:48

[QUOTE=garambois;556646]After a few checks on the data, I would note that :
[code]
29^15:i38 merges with 18528:i1
[/code]should be :
[code]
29^15:i37 merges with 18528:i0
[/code]Do you confirm that I should make this change ?[/QUOTE]
Again! What the Heck is wrong with my script?* Yes, I confirm the error! You should change it accordingly.

[SIZE=1]*Has to be something to do with index 0, obviously. . .[/SIZE]

 garambois 2020-09-10 18:43

OK, I'll make the change for this weekend's update.
Thanks Ed !
For this update, I won't be advanced enough in the calculations for base 50, I won't add it for the moment.
Base 72 can't be added either because of the malfunction.
For base 98, you can keep me informed if it is to be added this weekend or at a later date.
Many thanks.

:smile:

 RichD 2020-09-10 20:49

20^59

All of n=770 should be colored in. Releasing table.

 EdH 2020-09-11 02:22

[QUOTE=garambois;556655]OK, I'll make the change for this weekend's update.
Thanks Ed !
For this update, I won't be advanced enough in the calculations for base 50, I won't add it for the moment.
Base 72 can't be added either because of the malfunction.
For base 98, you can keep me informed if it is to be added this weekend or at a later date.
Many thanks.

:smile:[/QUOTE]Well, hopefully all this report is accurate:

Base 18 is green through 18^119 and orange for further by default.

Base 72 is green through 72^74 and orange beyond, but I will still need to post the .elfs for the corrupted sequences.

Base 98 is green through 98^71 and orange for further by default.

I have not yet completed index 1 for many of the orange table blocks.

 Happy5214 2020-09-11 11:19

Releasing 21^72, 21^74, and 21^76.

21^78 got down to 42 digits in the midst of three separate downdriver runs, but it's now at 117 digits with a 2^2*7 driver and a 3. :cry:

 garambois 2020-09-11 16:59

20^59

All of n=770 should be colored in. Releasing table.[/QUOTE]

OK, many thanks !
20^59 is a non-trivial end of sequence !!!

 garambois 2020-09-11 17:09

[QUOTE=EdH;556691]Well, hopefully all this report is accurate:

Base 18 is green through 18^119 and orange for further by default.

Base 72 is green through 72^74 and orange beyond, but I will still need to post the .elfs for the corrupted sequences.

Base 98 is green through 98^71 and orange for further by default.

I have not yet completed index 1 for many of the orange table blocks.[/QUOTE]

OK, at the next update, I will add the base 98.

And what do you think : is it worth adding base 72 ?
But several cells may present an absurd result ?

 garambois 2020-09-11 17:13

[QUOTE=Happy5214;556718]Releasing 21^72, 21^74, and 21^76.

21^78 got down to 42 digits in the midst of three separate downdriver runs, but it's now at 117 digits with a 2^2*7 driver and a 3. :cry:[/QUOTE]

Thanks a lot.
Pour 21^78, vous avez maintenant perdu le 3 !

 Happy5214 2020-09-11 17:23

[QUOTE=garambois;556742]Thanks a lot.
Pour 21^78, vous avez maintenant perdu le 3 ![/QUOTE]

English translation (I don't speak French): [i]For 21^78, you have now lost the 3![/i]

No, there are almost 100 terms I have yet to upload. I'll do that now. :wink:

Edit: They're up now. Look at the stinking 3. Look at it! :sad:

 garambois 2020-09-11 18:30

Sorry for the French sentence, it's a false manipulation !
Yes, indeed, the 3 is there !

 EdH 2020-09-11 20:48

1 Attachment(s)
[QUOTE=garambois;556741]OK, at the next update, I will add the base 98.

And what do you think : is it worth adding base 72 ?
But several cells may present an absurd result ?[/QUOTE]
That's up to you, but unless you are going to do the extra to supply links to valid sequences I'd recommend against adding the whole table. What about adding the table up through [I]i=52[/I] with a static note about the problem? The db has all the valid sequences and I'm attaching the correct ones for the ten that are invalid in the db. The top four of the attached ones are just factored through index 1. The rest are all the way to prime. So, any data retrieval can be accomplished.

 RichD 2020-09-12 01:11

Going into the weekend update, my only reservations would be for n=20 and i=75 through 91 with only odd exponent.

 garambois 2020-09-13 10:01

OK, page updated.
Thank you all for your contribution !
The speed of progress of the project is really phenomenal right now !

For base 72, I'm still waiting, hoping that the problem on factodb will be solved. If the problem is not solved, in a few weeks, I will add the base 72 up to i=52 by adding a note, as suggested by Edwin.

Thank you all to check again your information on the updated page, to see if there are no errors...

 EdH 2020-09-13 12:19

All looks good from my viewpoint. I do see that I missed factoring index 1 for 98^72. That will be finished in a little while, so it'll be available for any data gathering, even though the table won't show it.

I think I have index 1 finished for 72^75 through 72^88, either entered into the db or within my earlier attachment. I plan to enter index 1 and beyond for all the finished of the corrupted sequences into the db. That way, when they are fixed the whole sequence will be available there.

I intend to factor index 1 for 18^120 through 18^129 next, so those will be available for study.

 EdH 2020-09-13 13:45

[QUOTE=EdH;556850]All looks good from my viewpoint. I do see that I missed factoring index 1 for 98^72. That will be finished in a little while, so it'll be available for any data gathering, even though the table won't show it.[/QUOTE]98^72 index 1 is now factored.

[QUOTE=EdH;556850]I think I have index 1 finished for 72^75 through 72^88, either entered into the db or within my earlier attachment. I plan to enter index 1 and beyond for all the finished of the corrupted sequences into the db. That way, when they are fixed the whole sequence will be available there.[/QUOTE]All of the corrupted sequences have been updated in the db. Index 1 and beyond all appear to be correct.

[QUOTE=EdH;556850]I intend to factor index 1 for 18^120 through 18^129 next, so those will be available for study.[/QUOTE]This is in work.

@Jean-Luc: Thanks for all the work keeping the tables up to date. I'm open for suggestions as to where you might like me to look after I get done with the current work. It may be a few days, but I'm not sure where my interest may wander after that.

 garambois 2020-09-13 15:43

Edwin, thank you very much for your help !
We now have a lot of bases started.
I think the wisest thing to do is to try to green as many cells as possible. Especially sequences that end trivially. But as the calculations become more and more difficult, it may take months. After that, for my part, I will try to study the prime numbers that end the sequences. The goal will be to see if there is a link between the base and these prime numbers. I will let you know when I start this new analysis work.
In the meantime, I am finishing my calculations for bases 30, 50 and 9699690.

I may also devote some threads again to other ideas that have been put on hold for the benefit of this project's work. I have a lot of programs on hold, having only 12 threads on my CPU. I dream of a computer with 120 threads that would allow me to run all the programs I have written (work on cycles, on aliquot sequences with complex integer Gauss numbers, on the growth coefficients of successive terms of aliquot sequences, calculations of sequences of the main project, work on the lengths of the branches of the infinite graph of the aliquots sequences and many other ideas...). But I still have to wait 1 or 2 years, because a CPU with 120 threads has a certain price, as well as the amount of RAM needed and everything that goes with it !

:smile:

 Stargate38 2020-09-13 23:56

Looks like the problem is far worse than we thought:

[url]http://www.factordb.com/index.php?query=72^x&use=x&x=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1[/url]

If you keep hitting "Next Page > >", you'll see that it's happening to a lot of powers of 72.

 warachwe 2020-09-14 09:44

[QUOTE=Stargate38;556914]Looks like the problem is far worse than we thought:

[url]http://www.factordb.com/index.php?query=72^x&use=x&x=1&VP=on&VC=on&EV=on&OD=on&PR=on&FF=on&PRP=on&CF=on&U=on&C=on&perpage=200&format=1[/url]

If you keep hitting "Next Page > >", you'll see that it's happening to a lot of powers of 72.[/QUOTE]

The problem seem to affect only n that is divided by some prime bigger than 50. Somehow it change 72^n to 2^(6*n).

 RichD 2020-09-14 12:47

I see my notes are in error. I will "color" the cell 770^33 later this week. Thank you for the detailed and nice table layout.

 garambois 2020-09-14 19:07

@Stargate38 & warachwe :
I don't know what's going on ? I hope someone will be able to fix this problem ?
Fortunately, it doesn't seem that we have this problem for other bases !

@RichD : Thanks a lot !

 yoyo 2020-09-14 20:03

I'll take bases 14, 15, 17, 18, 19 as usual until the remaining composite is > C139.

 richs 2020-09-14 21:59

[QUOTE=richs;556259]Taking 439^38 at i52.[/QUOTE]

439^38 is now at i185 (added 133 iterations) and a C121 level with a 2^4 * 3^2 guide, so I will drop this reservation. The remaining C108 term is well ecm'ed and is ready for nfs.

Taking 439^20 at i2157.

 EdH 2020-09-14 22:25

[QUOTE=yoyo;556976]I'll take bases 14, 15, 17, 18, 19 as usual until the remaining composite is > C139.[/QUOTE]
I've terminated 18^120 and only three others of the rest are <C140 for their remaining cofactor. Perhaps you should take all of them to prime through 18^129.

 yoyo 2020-09-15 05:11

In fact I took all from 18^0 to 18^140. But many of them dropped out because they have either terminated or a composite > C139.

 RichD 2020-09-15 05:56

[QUOTE=yoyo;556976]I'll take bases 14, 15, 17, 18, 19 as usual until the remaining composite is > C139.[/QUOTE]

So few numbers are still available in base n=18. I will release my reservations for the few numbers I have in base 20. [B]yoyo[/B] can have the entire table. I will focus on 770^33 and begin preliminary work on Table 31. I believe this is the last unclaimed base in [url=https://www.mersenneforum.org/showpost.php?p=554359&postcount=457]post #457[/url].

 EdH 2020-09-15 12:40

[QUOTE=yoyo;557002]In fact I took all from 18^0 to 18^140. But many of them dropped out because they have either terminated or a composite > C139.[/QUOTE]
All the ones from 18^0 through 18^120 dropped out because RichD (<18^97) and I (>18^96) ran them to prime a while ago. But the ones left above exponent 120 were because they were >c140 for index 0. Three of those had cofactors <c140 and I see that they are now >c139. I was suggesting that you might want to take those 9 sequences to prime, even though they are currently above your >c139 threshold because they are all falling and will terminate soon.

Let me know, though, since if you do not want those 9, I will work on them leisurely.

 EdH 2020-09-15 12:53

[QUOTE=RichD;557003]So few numbers are still available in base n=18. I will release my reservations for the few numbers I have in base 20. [B]yoyo[/B] can have the entire table. I will focus on 770^33 and begin preliminary work on Table 31. I believe this is the last unclaimed base in [URL="https://www.mersenneforum.org/showpost.php?p=554359&postcount=457"]post #457[/URL].[/QUOTE]
There are still two unaddressed bases suggested in [URL="https://www.mersenneforum.org/showpost.php?p=547156&postcount=280"]post #280[/URL]: 220 and 284. But, I'm not sure of the interest in those (amicable) bases ATM.

 garambois 2020-09-15 15:59

OK, I will take into account all your booking requests when I update next weekend.

I'm also waiting to know who will finally book the base 18 ?
For the moment, if you don't give me more details about this base 18, I will book it for EDH with the goal to finish the 9 remaining sequences.

[QUOTE=EdH;557032]There are still two unaddressed bases suggested in [URL="https://www.mersenneforum.org/showpost.php?p=547156&postcount=280"]post #280[/URL]: 220 and 284. But, I'm not sure of the interest in those (amicable) bases ATM.[/QUOTE]

I am also not sure of the interest of calculating the sequences of the bases 220 and 284, two amicable numbers !
On the other hand, if we calculate them, and if we find that the sequences have no particular interest, at least we will be certain. But you never know: if there was something "special" going on with the bases that are amicable numbers, it would be too stupid to miss it. If someone wants to calculate these two bases, I will gladly add them to the page, after the preliminary work.

 yoyo 2020-09-15 16:17

I will currently not go above C139. You can book it to EdH.
For base 18 volunteers have only 18^126 and 18^138 running.

 EdH 2020-09-15 16:41

[QUOTE=yoyo;557054]I will currently not go above C139. You can book it to EdH.
For base 18 volunteers have only 18^126 and 18^138 running.[/QUOTE]
Thanks yoyo! I won't be near those for quite some time.

 EdH 2020-09-15 16:45

[QUOTE=garambois;557051]OK, I will take into account all your booking requests when I update next weekend.

I'm also waiting to know who will finally book the base 18 ?
For the moment, if you don't give me more details about this base 18, I will book it for EDH with the goal to finish the 9 remaining sequences.

I am also not sure of the interest of calculating the sequences of the bases 220 and 284, two amicable numbers !
On the other hand, if we calculate them, and if we find that the sequences have no particular interest, at least we will be certain. But you never know: if there was something "special" going on with the bases that are amicable numbers, it would be too stupid to miss it. If someone wants to calculate these two bases, I will gladly add them to the page, after the preliminary work.[/QUOTE]At least for the near term, I will leave 220 and 284 for those with less resources. I'll work the higher sequences for base 18 and maybe look at the higher ones for base 98 later on.

Thanks Jean-Luc!

 RichD 2020-09-15 16:56

Looks like 31^4 and 31^18 may have merged. Can someone verify this?

I am releasing all base 20 numbers.

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