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Old 2021-01-30, 17:58   #716
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Quote:
Originally Posted by Happy5214 View Post
. . ., plus 26^67, which had additional terms already in FactorDB that I didn't compute, but I couldn't deduce a merge for. . .
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.
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Old 2021-01-31, 00:01   #717
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Would you like table 338 ( 2 * 13^2 ) initialized?
Would base 37 be more of an interest?
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Old 2021-01-31, 00:49   #718
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Quote:
Originally Posted by EdH View Post
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.
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.
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Old 2021-01-31, 03:23   #719
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Originally Posted by Happy5214 View Post
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.
Yeah, I couldn't come up with any idea as to why it may have been run.
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Old 2021-01-31, 14:01   #720
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New orientation of the project ?

I) Prime numbers at the end of sequences and occurrences of prime numbers in the sequences

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 (https://www.mersenneforum.org/showth...12&page=41#447) 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.

II) A new direction for the project : cycles statistics

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.

III) Proposal for further calculations

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).

But I would like to make a final remark.

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 !
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Old 2021-01-31, 14:04   #721
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Quote:
Originally Posted by RichD View Post
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:
Originally Posted by RichD View Post
Would base 37 be more of an interest?

Yes, of course for both, following what was said in the previous post !
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Old 2021-01-31, 23:04   #722
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Some early observations from base 37. The following sequences may have merged.

37^6
37^8
37^12
37^22
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Old 2021-02-01, 00:00   #723
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Quote:
Originally Posted by RichD View Post
Some early observations from base 37. The following sequences may have merged.

37^6
37^8
37^12
37^22
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
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Old 2021-02-01, 16:57   #724
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OK, I find the same results for all 4 mergers.
Thank you.
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Old 2021-02-01, 19:16   #725
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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.
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Old 2021-02-01, 19:41   #726
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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 !
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