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

 VBCurtis 2020-11-22 18:18

That's not unusual, at all! I guess I just don't encounter 2^4 * 31 often enough to pay attention. I appreciate the data!

 garambois 2020-11-22 19:01

[QUOTE=VBCurtis;564022]13^64. It got down under 50 digits, but picked up 2 * 3 so it's back above 110 dig now.

13^58, 60, 62 can be updated on your page also.[/QUOTE]

Ok for 13^64, thanks...

The next update will take place next weekend.

 EdH 2020-11-23 22:05

[QUOTE=VBCurtis;563992]How unusual is it to break 2^4 * 31? I don't think I've done it before tonight.

Edit: The sequence went 2^4 * 3 * 31 to 2^4 * 3 * 31^2 to 2^6 * 3 * 31, and picked up the downdriver a few terms later.[/QUOTE]
My earlier numbers are probably quite inaccurate! I may need to focus on 2^4 only, rather than 2[SUP](>4)[/SUP]. I did a bit more study and it appears that 2[SUP]4[/SUP] * 31 is specifically important! If the power of either 2 or 31 changes, the driver has a really good chance of breaking. There are instances of 2[SUP](>4)[/SUP] * 31 and 2^4 * 31[SUP]2[/SUP] not starting a run.

Example for 10^27 (first column is index number):
[code]
52 = [B]2^7[/B] * 31 * 37 . . .
170 = [B]2^5[/B] * 3^2 * 31 * 41 . . .
687 = [B]2^6[/B] * 3 * 31^2 * 59 . . .
754 = 2^4 * 7 * 31 * 223 . . .
755 = 2^4 * 7 * 31 * . . .
. . . (2^4 continues for all these lines)
1081 = 2^4 * 3 * 13 * 31 * 2207 . . .
1082 = 2^4 * 3 * 5 * 31 * 47681 . . .
[/code]1082 is the current last term, so this instance is unbroken.

Another example 10^111:
[code]
592 = 2^4 * 31 * 73 . . .
593 = 2^4 * 31 * 1493 . . .
. . .
674 = 2^4 * 13 * 31 * 128311 . . .
675 = 2^4 * 5 * [B]31^2[/B] * 1407503 . . .
676 = 2^4 * 5^2 * [B]31^2[/B] * 89 . . .
677 = 2^4 * 5^2 * 17 * [B]31^2[/B] . . .
[/code]1087 is the current last term, so this instance broke at 677.

I will do even more study and post something extra later. . .

 EdH 2020-11-23 23:04

[QUOTE=EdH;564139]. . .
I will do even more study and post something extra later. . .[/QUOTE]
I reran the above using only 2[SUP]4[/SUP] and came up with 70 unbroken and 296 broken. This shouldn't be too far off. Some variations may not be considered, but the runs still appear to break much more often than not.

Perhaps I should look at other drivers/guides?

I can't seem to find the page which describes the difference between drivers and guides. Is it still available somewhere?

 VBCurtis 2020-11-23 23:34

Squaring the 31 is the manner in which one breaks 2^4 * 31.
Other powers of 2 would not be a driver; only the perfect number 496 = 2^4 * 31 is a driver.
I believe the same is true for the other perfect numbers: Square the 3 or 7 or 127 to escape.

 LaurV 2020-11-24 05:02

[QUOTE=VBCurtis;564151]I believe the same is true for the other perfect numbers: Square the 3 or 7 or 127 to escape.[/QUOTE]
Yep. There is a thread discussing the math, somewhere here around.

 richs 2020-11-25 04:49

[QUOTE=richs;561799]439^40 is now at i554 (added 515 lines) and a C122 level with a 2^2 * 5 * 7 driver, so I will drop this reservation. The remaining C116 term is well ecm'ed and is ready for nfs.

Taking 439^42 at i15 and 439^44 at i2.[/QUOTE]

439^42 is now at i65 (added 50 lines) and a C131 level with a 2^3 * 3 * 5 driver so I will drop this reservation. The remaining C126 term is well ecm'ed and is ready for GNFS.

Continuing with 439^44.

 garambois 2020-11-29 10:50

Page updated.
Many thanks to all for your help !

 unconnected 2020-12-06 15:03

I'm done with base 79 - all sequences in the table are >120 digits with >110 ECM-ed composites. Also I did some work with odd k's above the limit, will post results later.

 garambois 2020-12-06 15:50

[QUOTE=unconnected;565437]I'm done with base 79 - all sequences in the table are >120 digits with >110 ECM-ed composites.[/QUOTE]

OK, I will make the next update in one or two weeks.

[QUOTE=unconnected;565437]Also I did some work with odd k's above the limit, will post results later.[/QUOTE]

All right, keep me posted. I can always extend the base by adding more exponents.

Many thanks !

 RichD 2020-12-13 12:13

One of the next logical tables would be 242 (2 * 11^2). I've taken it to i=52.

Edit: Updates to tables 29, 220, 284.

 garambois 2020-12-13 15:02

[QUOTE=RichD;566074]One of the next logical tables would be 242 (2 * 11^2). I've taken it to i=52.

Edit: Updates to tables 29, 220, 284.[/QUOTE]

I will update all the bases on the weekend of December 20th.
And I will also add the base 242 that you calculated.
Thus, the base 200 = 2 * 10^2 will be missing, if you like the bases for which the calculations end for all the exponents. These bases will be interesting for future analyses.

 RichD 2020-12-13 17:43

[QUOTE=garambois;566087]Thus, the base 200 = 2 * 10^2 will be missing, if you like the bases for which the calculations end for all the exponents. These bases will be interesting for future analyses.[/QUOTE]

I'm on it.

 garambois 2020-12-13 17:52

OK, many thanks for your help !

 garambois 2020-12-20 09:26

OK, page updated.
Many thanks to all for your help.

Base 242 added up to exponent 52.

[QUOTE=unconnected;565437]Also I did some work with odd k's above the limit, will post results later.[/QUOTE]

Please, let me know when I can add the exponents for base 79. For the moment, I see that you have extended to exponent k=85.

 RichD 2020-12-20 15:55

[QUOTE=garambois;566756]Base 242 added up to exponent 52.[/QUOTE]

I took it to 54. :smile:

 RichD 2020-12-21 22:14

Oops, I mean Base 200 was taken to exponent 54.

 garambois 2020-12-21 23:18

OK. If I summarize :
Base 242 up to exponent 52 and base 200 up to exponent 54, right ?

 RichD 2020-12-22 01:16

[QUOTE=garambois;566925]OK. If I summarize :
Base 242 up to exponent 52 and base 200 up to exponent 54, right ?[/QUOTE]

Correct.

 RichD 2020-12-23 00:01

Base 288 (2 * 12^2) or (2^3 * 3 ) is ready for table insertion at the next update. Exponent to 50.

 garambois 2020-12-23 08:52

I will add these two additional bases in the next update. Thank you very much !

 RichD 2020-12-23 10:50

[QUOTE=RichD;567037]Base 288 (2 * 12^2) or (2^3 * 3 ) is ready for table insertion at the next update. Exponent to 50.[/QUOTE]

(2^5 * 3^2)
I can't do math. :smile:

 Happy5214 2020-12-25 08:01

Apologies for poaching from yoyo, but with a little work with FactorDB assisted by the [URL="https://www.alpertron.com.ar/ECM.HTM"]Alpertron ECM/SIQS tool[/URL], I found that 10^129:i2229 merges with 2173380:i159.

 garambois 2020-12-25 08:49

[QUOTE=Happy5214;567288]Apologies for poaching from yoyo, but with a little work with FactorDB assisted by the [URL="https://www.alpertron.com.ar/ECM.HTM"]Alpertron ECM/SIQS tool[/URL], I found that 10^129:i2229 merges with 2173380:i159.[/QUOTE]

That's right !
Thank you very much !
I will make an update on December 26th or 27th that will take into account the latest messages !

 richs 2020-12-25 14:42

439^44

[QUOTE=richs;564297]Continuing with 439^44.[/QUOTE]

439^44 is now at i98 (added 96 lines) and a C126 level with a 2^2 * 7 guide so I will drop this reservation. The remaining C117 term is well ecm'ed and is ready for GNFS.

This completes the 439 table to the C120 level or termination.

Merry Christmas to all!

 garambois 2020-12-25 18:30

OK, base 439 complete.
A lot of thanks.
Merry Christmas to you too and to all of you !

 garambois 2020-12-27 10:37

OK, page updated.
Many thanks to all for your contribution !

Base 200 added up to exponent 54.
Base 288 added up to exponent 50.
Base 439, work completed.
Addition of exponents 64 to 95 for base 79.

I still have to correct some attributions for base 288, at the next update...

There are now a total of 45 bases on the page !

:smile:

 EdH 2020-12-27 16:04

[QUOTE=garambois;567440]. . .
There are now a total of 45 bases on the page !

:smile:[/QUOTE]And, if my count is correct, 4618 sequences. . .

 garambois 2020-12-30 09:16

Your addition is correct, I find the same. And I also added the other columns, it gives :

All : 4618
Done : 2986
Open : 1608
Merges : 78
Cycle : 24

And we have the exact addition : 24 + 1608 + 2986 = 4618

Note that 24 + 2986 = 3010.
So we have 3010 / 4618 = 0.6518 or about 65% of all our aliquot sequences for which we have finished the caculs !

 RichD 2020-12-30 10:59

29^36 terminates.

 garambois 2020-12-30 19:00

Waouh : a non-trivial ending !
Congratulations !

29^15 is also to my knowledge the only aliquot sequence in our project that should end trivially and that doesn't ! At index 36, we find a perfect square term. So we have a change of parity. And we have at index 37 the number 18528 which is the start of an Open-End sequence.

 garambois 2021-01-17 20:04

OK, page updated.
Many thanks to all for your help !

[U]Notable discovery for this time :[/U]
The sequence 3^286 ends with the 28-cycle.
- This is the first sequence of the project that ends on the 28-cycle !
- And it is the first sequence that ends on a cycle for base 3 !
- And we hadn't had any sequence that ended with a cycle of length other than 1 or 2 before that in this project !
Karsten Bonath had flair in pushing further the calculations for this sequence !
Congratulations !

Now, let's wait to find another sequence that ends with a cycle for base 3 to see how long its cycle length will be.

:smile:

 Happy5214 2021-01-29 21:27

After a little more than 4,000 terms and a heart-stopping roller-coaster ride in the middle, my run with 24^21 ends with nothing to show for it.

There are a couple of glaring holes in the list, bases 22 and 26. I'd like to initialize those to 100 digits. The even exponents are already done for both up to 120 digits.

Also, would it be too much trouble to generate a page like [url]http://www.aliquotes.com/OE_3000000_C80.txt[/url] for the power sequences?

 Happy5214 2021-01-30 07:23

Base 26 is done to 100 digits (26^69). There were 3 merges (26^15:i350 = 2360:i4, 26^35:i503 = 87612:i4, and 26^55:i458 = 871000:i0), plus 26^67, which had additional terms already in FactorDB that I didn't compute, but I couldn't deduce a merge for. Exponents 1, 3, 5, 19, 31, 33, and 59 terminate. I'll start base 22 next.

 garambois 2021-01-30 15:36

OK, page updated.
Thanks to all for your help !

[QUOTE=Happy5214;570449]After a little more than 4,000 terms and a heart-stopping roller-coaster ride in the middle, my run with 24^21 ends with nothing to show for it.
[/QUOTE]

OK, thanks !

[QUOTE=Happy5214;570449]
There are a couple of glaring holes in the list, bases 22 and 26. I'd like to initialize those to 100 digits. The even exponents are already done for both up to 120 digits.
...
...
Base 26 is done to 100 digits (26^69). There were 3 merges (26^15:i350 = 2360:i4, 26^35:i503 = 87612:i4, and 26^55:i458 = 871000:i0), plus 26^67, which had additional terms already in FactorDB that I didn't compute, but I couldn't deduce a merge for. Exponents 1, 3, 5, 19, 31, 33, and 59 terminate.[/QUOTE]

OK, thanks !
Congrats on the non-trivial sequences that are coming to an end !
For mergers, I find exactly the same as you.
For 26^67, sorry, but I can't find better than you can of which merger it could be ?
I am waiting to hear from you for base 22.

Yes, there are still holes in the list. At the beginning of the project, we wanted to calculate all the bases up to 100, but when we realized the difficulty of the task, we revised our ambitions downwards !

[QUOTE=Happy5214;570449]
Also, would it be too much trouble to generate a page like [URL]http://www.aliquotes.com/OE_3000000_C80.txt[/URL] for the power sequences?[/QUOTE]

This is a good idea.
I will think about it during my next vacations in a few weeks.

:smile:

 RichD 2021-01-30 17:35

Would you like table 338 ( 2 * 13^2 ) initialized? Or do you have enough data for 2 * prime^2 to understand the trend? I can start on it if you like.

 EdH 2021-01-30 17:58

[QUOTE=Happy5214;570486]. . ., plus 26^67, which had additional terms already in FactorDB that I didn't compute, but I couldn't deduce a merge for. . .[/QUOTE]Looking at the "More information" section for all the terms, it looks like the entire sequence was already in the db. They seem to all show, "Before November 4, 2018, 12:20 am" for the Create times. If this is the case, perhaps there is no merge.

 RichD 2021-01-31 00:01

[QUOTE=RichD;570511]Would you like table 338 ( 2 * 13^2 ) initialized?[/QUOTE]

Would base 37 be more of an interest?

 Happy5214 2021-01-31 00:49

[QUOTE=EdH;570513]Looking at the "More information" section for all the terms, it looks like the entire sequence was already in the db. They seem to all show, "Before November 4, 2018, 12:20 am" for the Create times. If this is the case, perhaps there is no merge.[/QUOTE]

Thanks for the detective work! I just looked at the aliqueit log for that sequence, and it's basically empty, which confirms the prior existence of those terms. It seems like a rather random sequence and depth though.

 EdH 2021-01-31 03:23

[QUOTE=Happy5214;570537]Thanks for the detective work! I just looked at the aliqueit log for that sequence, and it's basically empty, which confirms the prior existence of those terms. It seems like a rather random sequence and depth though.[/QUOTE]Yeah, I couldn't come up with any idea as to why it may have been run.

 garambois 2021-01-31 14:01

[SIZE=5]New orientation of the project ?[/SIZE]

[B]I) Prime numbers at the end of sequences and occurrences of prime numbers in the sequences[/B]

The great work on the occurrences of prime numbers in the sequences and on the prime numbers that end the sequences will be continued next summer.
Indeed, the data analysis work done last summer that led to the statement of the 133 conjectures ([URL]https://www.mersenneforum.org/showthread.php?t=23612&page=41#447[/URL]) must be continued with a tenfold increase in the amount of data.
And this is possible thanks to the work of everyone and of the yafu project, which performs an incredible amount of calculations.

[B]II) A new direction for the project : cycles statistics[/B]

The amount of data available has increased so much since the beginning of yafu's involvement that it becomes possible to approach a new type of work.
So, let me present a few things that I have noticed about the end of sequences on cycles :

1) All the ends of sequences on C1 end in the perfect number 6 except 28, 496, 8128 with the exponent 1 (the three other bases which are perfect numbers).
2) All sequences ending on C2 end on a different pair of amicable numbers, except 220^1 and 284^1, both belonging to the same amicable pair.
3) The only bases where the sequences ending on cycles of two different lengths C1 and C2 at the same time are bases that are perfect numbers (6 exponents 1 and 55 and 8128 exponents 1 and 11).

All these remarks must be coincidences!
We can prove that they are only coincidences by finding for example a base which would not be a perfect number and for which sequences would lead to cycles of different lengths.
But it becomes very difficult to find other cycles in the bases that we have calculated for the moment. It is very unlikely that we will be able to find other sequences ending in a cycle for bases 2, 3... But we have to try.
We only have 25 sequences in the whole project that end in a cycle and that's too few.

[B]III) Proposal for further calculations[/B]

To succeed in noticing more relevant things and to avoid going in the wrong direction when looking at the cycles which are ends of sequences, it would therefore probably be interesting to continue the calculations as proposed below, for people who want to do calculations for this project :

1) Continue the work on the sequences already started in the project (especially the small bases 2, 3, up to 30) and in this, the work of yafu is fundamental. It is then up to us to analyse the yafu finds in order to possibly push further the calculations for certain sequences.
2) Complete all the green columns for all the bases to note which sequences should end trivially and which do not (there is only one for the moment : 29^15).
3) Calculate new bases, if possible at least all the bases up to 50 exhaustively (to know how many new sequences fall on cycles).
4) Compute as much as possible new bases of the type 2*n^2, at least for n = 13, 14, 15, 17, 18, 19, 20, 21, 22 (to find many new sequences falling on cycles and OpenEnd sequences which must be very rare for this type of base).
5) Calculate base 33550336 (5th perfect number) and if possible continue the calculations for bases 28 and 496 (to find other cycles and see if they have a length different from 1, as is the case for 6 and 8128).

[B]But I would like to make a final remark.[/B]

I'm well aware that this project mobilizes a great deal of computing power, since we also have yafu working on it.
The work of the main project (OpenEnd suites from 1 to 3*10^6) is therefore slowed down because of this project on suites that start on integer powers.
However, everyone must understand that we are not at all sure that we will succeed in state new conjectures by analyzing the data.
Because the fundamental problem is that we don't know exactly what to look for. The ideas behind the project did not materialize as expected.
And we didn't expect this to happen at all : everything went in a different direction from what we had originally planned.
So we say to anyone who spends time on this project and uses electricity for it : only do this if you find pleasure and personal interest in it !
We are not sure we will have a result that meets our expectations !

 garambois 2021-01-31 14:04

[QUOTE=RichD;570511]Would you like table 338 ( 2 * 13^2 ) initialized? Or do you have enough data for 2 * prime^2 to understand the trend? I can start on it if you like.[/QUOTE]

[QUOTE=RichD;570532]Would base 37 be more of an interest?[/QUOTE]

Yes, of course for both, following what was said in the previous post !

 RichD 2021-01-31 23:04

Some early observations from base 37. The following sequences may have merged.

37^6
37^8
37^12
37^22

 EdH 2021-02-01 00:00

[QUOTE=RichD;570613]Some early observations from base 37. The following sequences may have merged.

37^6
37^8
37^12
37^22[/QUOTE]Verified:
[code]
37^6:i126 merges with 109920:i1275
37^8:i151 merges with 1632:i37
37^12:i1057 merges with 10824:i28
37^22:i519 merges with 1567300:i0
[/code]

 garambois 2021-02-01 16:57

OK, I find the same results for all 4 mergers.
Thank you.

 EdH 2021-02-01 19:16

I will work on the table for 33550336, but it will still be a few days before I start, while I complete some other things.

 garambois 2021-02-01 19:41

Many thanks to all !

I can't wait to see if we will have cycles for bases 22, 37 and 33550336 which will be added very soon.

For base 3, I'm not deluding myself too much : it is very unlikely that we will find another cycle. But the 28-length cycle for the 3^286 sequence is exceptional in more ways than one ! In general, we have cycles for not too large exponents, but it is different here !

 Happy5214 2021-02-02 09:36

[QUOTE=Happy5214;570486]... plus 26^67, which had additional terms already in FactorDB that I didn't compute, but I couldn't deduce a merge for. ...[/QUOTE]

One quick note from base 22, which will finish initializing either later Tuesday or Wednesday, is that 22^67 was also already done to 111 digits. I looked back, and 24^67 was also advanced to a similar depth before I got there, so someone must have made a concerted effort on those [I]i[/I]=67 sequences.

 garambois 2021-02-02 17:27

[QUOTE=Happy5214;570709]One quick note from base 22, which will finish initializing either later Tuesday or Wednesday, is that 22^67 was also already done to 111 digits. I looked back, and 24^67 was also advanced to a similar depth before I got there, so someone must have made a concerted effort on those [I]i[/I]=67 sequences.[/QUOTE]

This is indeed very curious !
It would seem that for bases 33, 34, 35 and 37 too, calculations were done further for exponent 67.

 yoyo 2021-02-02 20:58

I'll take base 26 and 29.

 RichD 2021-02-03 03:03

Do we have another merge with 37^30 ?

 Happy5214 2021-02-03 04:24

Base 22 has been initialized. There were three merges (22^5:i19=86388:i4, 22^29:i321=5208:i6, and 22^41:i1065=14676:i15). Exponents 1, 3, 7, 11, 19, 21, and 23 terminate, with 22^3 terminating with a perfect number (6).

In unrelated news, I poached several yafu@home sequences with downdrivers (apologies) and terminated 20^71 and 21^70.

 garambois 2021-02-03 08:16

[QUOTE=yoyo;570762]I'll take base 26 and 29.[/QUOTE]

Many thanks !
Next weekend I will do the next update and it will be noted in the tables.

 garambois 2021-02-03 08:23

[QUOTE=RichD;570776]Do we have another merge with 37^30 ?[/QUOTE]

I find the following fusion :
37^30:i1193 with 35856:i3
Many thanks.

 garambois 2021-02-03 08:41

[QUOTE=Happy5214;570779]Base 22 has been initialized. There were three merges (22^5:i19=86388:i4, 22^29:i321=5208:i6, and 22^41:i1065=14676:i15). Exponents 1, 3, 7, 11, 19, 21, and 23 terminate, with 22^3 terminating with a perfect number (6).

In unrelated news, I poached several yafu@home sequences with downdrivers (apologies) and terminated 20^71 and 21^70.[/QUOTE]

Many thanks.
I confirm the accuracy of the mergers.
And so we have one more cycle, which is C1 = 6.
Base 22 will be added in the next update.

Yafu is an extremely powerful tool precisely to allow us to locate the sequences that must be calculated further. Without yafu, it would take months, if not years, to spot these sequences.
[B]We cannot thank yoyo enough for his entry into the project ![/B]

 yoyo 2021-02-03 16:10

I would take more, but only complete bases.
Which ones are important?
For the while beeing I'll take also base 31 and 33, they seems to be completly unreserved.

 garambois 2021-02-03 16:53

[QUOTE=yoyo;570795]I would take more, but only complete bases.
Which ones are important?
For the while beeing I'll take also base 31 and 33, they seems to be completly unreserved.[/QUOTE]

You mean bases 30 and 31 (and not 31 and 33), because base 33 has not yet started ?

And if you want to take even more, would it be possible to take in priority the perfect number 8128 and the two amicable numbers 220 and 284 ?

Note : In 2 weeks, I will be on vacation and I will take a closer look at the sequences that end with cycles. It is likely that we will have to add additional exponents for the perfect numbers 28, 496 and 8128, we will see...

 yoyo 2021-02-03 17:10

Oh, I took 33.
But I'll take 30, 8128, 220, 284 also.

 EdH 2021-02-03 17:34

If you guys would like, I can let yoyo have 33550336, as well. I have done very little with it so far. I'm really only playing with a Colab session with it ATM.

 garambois 2021-02-04 17:23

[QUOTE=yoyo;570798]Oh, I took 33.
But I'll take 30, 8128, 220, 284 also.[/QUOTE]

OK, thank you very much, this is great news ! Update next weekend.

 garambois 2021-02-04 17:29

[QUOTE=EdH;570800]If you guys would like, I can let yoyo have 33550336, as well. I have done very little with it so far. I'm really only playing with a Colab session with it ATM.[/QUOTE]

Don't worry, I'll do the preliminary work for the perfect number 33550336 myself within three weeks if you don't tell us on this forum that you're done by then.

 EdH 2021-02-04 17:57

[QUOTE=garambois;570866]Don't worry, I'll do the preliminary work for the perfect number 33550336 myself within three weeks if you don't tell us on this forum that you're done by then.[/QUOTE]I can take care of it. I've already done a little. I was just offering it to yoyo, in case he needed more to add to his project.

 Happy5214 2021-02-06 03:36

On the base summary lines, the fact that Done doesn't include sequences ending in cycles is confusing, since these sequences are in fact done. There are two possible solutions. We could include cycles in the Done count (a similar duplication to the merges being included with the open sequence count) or rename Done to Prime (the actual meaning of that color and parallel to Cycle). If we do the latter, we may want to reorder the stats so that Prime and Cycle are next to each other, perhaps duplicating the order in the key?

-----------

PS @yoyo You can also take base 22 if you want it. It will be added to the page in the next update, as Jean-Luc said earlier, and everything is at least 100 digits. I don't know if you missed that post, which includes the skipped sequences. 22^89 is 120 digits if you need the limiting reference.

 yoyo 2021-02-06 06:30

Oh, yes I missed that post. I just looked at the reservation page. I take base 22.

 garambois 2021-02-06 09:18

[QUOTE=Happy5214;570966]On the base summary lines, the fact that Done doesn't include sequences ending in cycles is confusing, since these sequences are in fact done. There are two possible solutions. We could include cycles in the Done count (a similar duplication to the merges being included with the open sequence count) or rename Done to Prime (the actual meaning of that color and parallel to Cycle). If we do the latter, we may want to reorder the stats so that Prime and Cycle are next to each other, perhaps duplicating the order in the key?
[/QUOTE]

I think you're right. For clarity, we have to rename "done" to "Prime" and change the order. If we make the following change, does that seem clearer to you ?

Replace this (base 2 example) :

[COLOR=Blue]Stats: All: 559 / Done: 556 ➔ 99.46% / Open: 0 ( 0) ➔ 0.00% / Merges: 0 ➔ 0.00% / Cycle: 3 ➔ 0.54%[/COLOR]

By this :

[COLOR=blue]Stats: All: 559 / Prime: 556 ➔ 99.46% / Cycle: 3 ➔ 0.54% / Open: 0 ( 0) ➔ 0.00% / Merges: 0 ➔ 0.00%[/COLOR]

 Happy5214 2021-02-06 12:29

[QUOTE=garambois;570970]I think you're right. For clarity, we have to rename "done" to "Prime" and change the order. If we make the following change, does that seem clearer to you ?[/QUOTE]

Much better.

 RichD 2021-02-06 22:44

Base 338 was taken to exponent 50. All showing green, as expected.

 garambois 2021-02-07 11:04

Page updated.
Many thanks to all.
Long and complicated update. So thank you all for checking to see if the updates correspond to your requests.

Bases 22, 33 and 338 added.
Bases 22, 26, 29, 30, 31, 33, 220, 284, 8128 reserved for yoyo.

For bases 22 and 33, I did not know up to which exponent the calculations had been made. So I looked on FactorDB up to which exponent calculations had been performed.

 EdH 2021-02-09 14:19

[QUOTE=garambois;571075]Page updated.
Many thanks to all.
Long and complicated update. So thank you all for checking to see if the updates correspond to your requests.

Bases 22, 33 and 338 added.
Bases 22, 26, 29, 30, 31, 33, 220, 284, 8128 reserved for yoyo.

For bases 22 and 33, I did not know up to which exponent the calculations had been made. So I looked on FactorDB up to which exponent calculations had been performed.[/QUOTE]Bummer - I just missed it! 33550336 base table is initialized through [I]i=20[/I]. Of interest (to me) was that [I]i=13[/I] started and ran for over 100 iterations hovering between 98 and 99 digits before moving on.

 garambois 2021-02-09 17:49

OK, thanks a lot.
I will add the 33550336 base in the next update.

 EdH 2021-02-09 19:09

Is anyone working with base 392 (2*14^2)? factordb does not seem to show any activity yet. I will start [STRIKE]initialization[/STRIKE] filling it in.

BTW, 392^17 ends with 6 (Cycle)

 garambois 2021-02-09 21:17

[QUOTE=EdH;571226]Is anyone working with base 392 (2*14^2)? factordb does not seem to show any activity yet. I will start [STRIKE]initialization[/STRIKE] filling it in.

BTW, 392^17 ends with 6 (Cycle)[/QUOTE]

To my knowledge, no one works on base 392. Thank you for working on this base.

 RichD 2021-02-11 18:52

Base 37 will be ready to insert into the tables at the next update. I will continue to work on it part-time. It was taken to exponent 80.

 EdH 2021-02-12 15:13

450 (2*15^2) appeared to be undisturbed until my inquiries, so I will start filling in the cells for a base 450 table. . .

 RichD 2021-02-12 17:47

Another merge 37^18 ?

 EdH 2021-02-12 19:25

[QUOTE=RichD;571442]Another merge 37^18 ?[/QUOTE]
Verified:
[code]
37^18:i1430 merges with 3366:i2
[/code]

 garambois 2021-02-12 20:20

OK, many thanks.

I will add the 37 base in the next update.
Keep me posted for bases 392 and 450.

I also found the following mergers :
33^8:i275 merges with 48024:i10
and
33^10:i94 merges with 1009656:i2
I will add these two mergers in the next update.

 EdH 2021-02-13 01:30

[QUOTE=garambois;571456]. . .
Keep me posted for bases 392 and 450.
. . .
[/QUOTE]Base 392 is all green up through [I]i=52[/I] (except for [I]i=17[/I], which cycles at 6).

 yoyo 2021-02-13 13:02

I need more food :-) roughly 200 sequences. I'll wait for updated pages to check which are the next bases.

 EdH 2021-02-13 15:37

[QUOTE=yoyo;571519]I need more food :-) roughly 200 sequences. I'll wait for updated pages to check which are the next bases.[/QUOTE]
What about the bases of the 6th (33550336) and 7th (8589869056) perfect numbers (or even further)?

The even parities will all terminate rather quickly, but most of the odd should give some good sequences. And you might turn up a cycle or two for the collection.

 yoyo 2021-02-13 15:54

Ok, I'll take both bases.

 EdH 2021-02-13 16:07

[QUOTE=yoyo;571528]Ok, I'll take both bases.[/QUOTE]
I also see that 496 still has quite a few odd parity sequences that haven't reached 140 digits. They don't show reservations and are on garambois' list.

 garambois 2021-02-13 16:18

[QUOTE=EdH;571525]What about the bases of the 6th (33550336) and 7th (8589869056) perfect numbers (or even further)?

The even parities will all terminate rather quickly, but most of the odd should give some good sequences. And you might turn up a cycle or two for the collection.[/QUOTE]

Absolutely yes ! Perfect numbers play a key role in our project !

[QUOTE=yoyo;571519]I need more food :-) roughly 200 sequences. I'll wait for updated pages to check which are the next bases.[/QUOTE]

This is really incredibly fantastic !!!

:smile:

If you really need more sequences, you can initialize bases 34 and 35 up to the minimum exponent 80 (or even 90). Because it is with these small bases that we are most likely to disprove the theory about cycles that we believe to be wrong exposed here :
If you want to calculate these two new bases, let me know. I would then add them to the page during a future update, when all the sequences whose exponents have the same parity as the base will end with a prime number. So the work is less laborious for me ;-).

Another solution : you can reserve bases already initialized from the project page and not yet calculated "very far" : bases 439, 770, 1155.
Or wait for tomorrow's update (February 14) to see other bases for which we can push the calculations further.

 yoyo 2021-02-13 16:19

Base 496 Taken.

 yoyo 2021-02-13 16:24

And take them all, base 34, 35, 439, 770, 1155 and will see how much it will be.

 garambois 2021-02-13 16:33

[QUOTE=EdH;571529]I also see that 496 still has quite a few odd parity sequences that haven't reached 140 digits. They don't show reservations and are on garambois' list.[/QUOTE]

Yes !

And I think we will also be forced to add exponents for base 28, up to 100 in a while, because it would be difficult to add exponents for larger perfect numbers, we would have too large integers too quickly !

 RichD 2021-02-13 16:34

I am starting the preliminary work on base 41 and base 43.

 garambois 2021-02-13 16:44

[QUOTE=yoyo;571528]Ok, I'll take both bases.[/QUOTE]

[QUOTE=yoyo;571531]Base 496 Taken.[/QUOTE]

[QUOTE=yoyo;571532]And take them all, base 34, 35, 439, 770, 1155 and will see how much it will be.[/QUOTE]

I think all of these bookings are going to represent a lot more than 200 sequences !
But a thousand times so much the better if yafu can do all this work !?
Thank you very much !

 garambois 2021-02-13 16:49

[QUOTE=RichD;571534]I am starting the preliminary work on base 41 and base 43.[/QUOTE]

Wonderful ! Today we have an explosion of activity even bigger than the other times... Thank you very much !

 EdH 2021-02-13 23:16

I'm nearing completion of base 450 through [I]i=50[/I], but three exponents (47, 49, 50) are unfinished.

I'll start work on 578 (2*17^2) next.

 Happy5214 2021-02-14 00:48

I just completed 882 (2*21^2) up to 120 digits (882^40). All terminated with primes except with 882^29, which terminated with a cycle (6)

I'm also finished with 24^27, and I'm releasing everything on base 24 below and including that.

If Karsten doesn't mind, I'd like to have a look at his page generation script. There are still some lingering HTML issues I want to take a crack at fixing.

 RichD 2021-02-14 01:03

Another merge 41^32 ?

 Happy5214 2021-02-14 01:17

[QUOTE=RichD;571562]Another merge 41^32 ?

41^32:i386 = 271980:i46

Edit: I'll also take base 42 to initialize.

 RichD 2021-02-14 04:20

Base 41 was taken to i=77 as will base 43 be taken to the same.

 Happy5214 2021-02-14 08:58

Base 42 done to 80 digits (odd exponents; 42^47) and 120 digits (even exponents; 42^72), and as soon as I wrote this, I think I misread [url]https://www.mersenneforum.org/showpost.php?p=571530&postcount=762[/url] above in reference to there being a new minimum. I wasn't planning to take this base to work on past the seeding work, so I could take it to the usual 100 digits or yoyo (or someone) could take it from here.

Exponents 1, 3, and 5 terminate with primes. There were two merges: 42^11:i131=35320:i2 and 42^17:i361=239940:i3. All even exponents tested terminated with primes, including 42^34, which terminated on the first term. Interestingly, 42^16 picked up a square and became even for two terms, before picking up another square and terminating two terms later.

 garambois 2021-02-14 12:08

Page updated.
Thank you all for your precious help !

[B]Added bases :[/B] 37, 41, 392, 450, 882 and 33550336.
[B]Bases reserved for yoyo :[/B] 439, 496, 770, 1155, 33550336.
[B]Bases being initialized which will be added during a next update, I will wait for your signal :[/B] 34 (YOY), 35 (YOY), 42 (YOY), 43 (RFD), 578 (EDH), 8589869056 (YOY).

Edit : When I look at the tables for bases 37 and 41, I think we forgot some merges, especially for base 37 ! But sorry, I really don't have enough time today to take care of this ! I will see for the next update ...

 yoyo 2021-02-14 12:21

I'll take also 37, 41 and 42.

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