20210528, 00:40  #1123  
"Alexander"
Nov 2008
The Alamo City
3×13×19 Posts 
Quote:


20210528, 10:05  #1124 
"Alexander"
Nov 2008
The Alamo City
2E5_{16} Posts 
Unrelated, 38^77 terminates at 59.

20210529, 00:38  #1125 
Sep 2008
Kansas
2×17×101 Posts 
Begin prelim work on base 74.

20210529, 01:51  #1126 
"Alexander"
Nov 2008
The Alamo City
1345_{8} Posts 
Doing preliminary work on bases 39 and 40 to 100 digits.

20210530, 08:25  #1127  
"Garambois JeanLuc"
Oct 2011
France
677 Posts 
Quote:
Many thanks ! We will see and we will keep you informed... 

20210530, 08:35  #1128  
"Garambois JeanLuc"
Oct 2011
France
1010100101_{2} Posts 
Page updated.
Thanks to all for your help ! Quote:
Many thanks, done. Many thanks. I look forward to hearing from you on this matter. Let me know when you have completed this preliminary work. I will then add these three new bases. Please, I remind you that for bases < 100, we do the calculations up to the exponent 100 to harmonize all the bases. Last fiddled with by garambois on 20210530 at 08:56 Reason: Added sentence about largest exponent for bases < 100 

20210530, 10:00  #1129 
"Garambois JeanLuc"
Oct 2011
France
677 Posts 
What can we do now ?
Let's now try to give an update on the status of our project. I remind you that today, we have three main lines of research for our project : 1) Work on the occurrences of prime numbers in the sequences and work on the prime numbers that terminate the sequences. 2) Work on the cycles that end sequences, according to the bases. 3) Work on the increasing beginnings of sequences ending in a trivial way. I don't know if a new line of research will be born in the near future, because in general, it happens randomly when we notice "something" interesting when we hadn't planned it at all. To try to see how to continue with our project, the easiest way is to set the objectives according to the three research axes mentioned above. Personally, intuitively, I think that we should go in the direction that spontaneously, without our consulting each other, some of you have started to follow :  Start by finishing calculating a maximum of trivial sequences added following the extension of many bases. This will be very useful for research axes 1) and 2).  Start new bases : we would certainly see more clearly if we had all the bases up to 100 and all the double bases of squares up to 1058 = 2 * 23^2. This will be very useful also for axes 1) and 2).  Try for all bases to go at least to index 2 or 3 for all exponents. This would allow to work not only on the abundance of s(n), but also on the abundance of s(s(n)) or further. So this would be very useful for axis 3). But I recognize that the calculations become very long for some bases. It's always the same problem when working on aliquot sequences ! As for adding bases belonging to cycles, I'm not sure what to think. Nothing special seems to happen with such bases. Maybe one of you will want to try the 14316 (28cycle) or 1264460 (4cycle) base or another... And now that we can sort and classify the bases of our project thanks to Karsten Bonath's new page, it is easy to see for example that we have only one base with 3 sequences that end in cycles, 4 bases with 2 sequences that end in cycles and 22 bases with one sequence that ends in a cycle. It will therefore be very difficult, if not impossible, to advance the research for axis 2) at this time. That said, I remain convinced that it is with bases < 10 that we have the best chance of advancing this research (axis 2). Maybe we will have to extend these bases in a few years, when our computers will be faster... It is in August that I will do the next big work of analysis of the bases. We'll see what comes out of it. Maybe unexpected things again ? Until then, let's color as many cells as possible in green or even better : in blue ! Perhaps some of you also have an intuition of "something" we should look at. Don't hesitate to propose ideas, this is our project. It is almost certain that there are many wonderful things hidden in the data we already have. We just lack the ideas to look at them. Thank you all for your efforts. 
20210601, 07:03  #1130 
"Alexander"
Nov 2008
The Alamo City
2E5_{16} Posts 
Base 39 done to 100 digits (trivial sequences terminated, all others taken to 100 digits or to termination/merge). i=2, 6, and 12 terminate with primes, and 39^4 merges:
Code:
39^4:i67 merges with 3876:i5 What does that entail for me? Do I have to do something above 39^63 (where I stopped) to finish the preliminary work for this base? In the past, I've completed the trivial sequences up to 120 digits, something I neglected to do here and probably could do in less than a day if you wanted me to. 
20210601, 08:28  #1131  
"Alexander"
Nov 2008
The Alamo City
2E5_{16} Posts 
Quote:
Last fiddled with by Happy5214 on 20210601 at 08:28 Reason: Removing extraneous syntax 

20210602, 09:56  #1132  
"Garambois JeanLuc"
Oct 2011
France
677 Posts 
Quote:
This is already special, since we have so few. But what would be really special is if there were another exponent of 14316 that ended in a cycle of length other than 1. This is what we are looking for, since so far this has only happened for bases that are perfect numbers. And this must really be due to pure chance ! But I must admit that I didn't look at 14316. 

20210602, 10:24  #1133  
"Garambois JeanLuc"
Oct 2011
France
677 Posts 
Quote:
Thank you very much for doing the preliminary work up to exponent 63. That's exactly what we needed to do. The reason I mentioned exponent 100 is because yoyo routinely runs all calculations up to exponent 100. So when I add a new base < 100 on our project page, I now routinely set the exponent to 100. For higher bases, I put exponents smaller and smaller as the base becomes larger. Anyway, even yoyo will never do calculations for terms larger than 150160 digits. And I understand that many people do calculations for our project, because the size of the numbers is still not too big compared to the main project. But after the preliminary work, some of us will reserve the base and may want to finish some trivial exponents for some bases and indeed, also to take the term size up to 120 digits for the openends sequences, or even 140 digits. I also thought that some people with a lot of computing power might want to push the work a bit further even for the preliminary phase, for example for exponents that end trivially. That's why I recalled that the exponents go to 100 for bases < 100. If you feel like pushing the preliminary work a bit further, but only if you find pleasure in it, that's our rule, I think it's better to act on sequences that end trivially than openends. And this is in view of the future analyses in August. But maybe other people see it differently and will ask the opposite... 

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