[QUOTE=garambois;552552]@EdH : I think it would be extremely efficient to generate the tables for the different bases by making only the prime numbers >= 10^4 appear and which in addition to that, also appear at several indexes in the same sequence !
I'll be able to do this myself in a few days, but maybe for you it's not too complicated and I'll see the results a few days in advance... No problem for me if you want to stop now and not generate these new tables, because all this is really a lot of work and requires really a lot of time ! :smile:[/QUOTE] I can very easily run everything again for all primes > 10^4, but as to limiting the lists to only those that show a repetition within a sequence, it might take a bit more work. Let me think about this a bit. In the meantime, I have decided to color in the transparent cells for base 2310 after all. 
[QUOTE=garambois;552552]@EdH : I think it would be extremely efficient to generate the tables for the different bases by making only the prime numbers >= 10^4 appear and which in addition to that, also appear at several indexes in the same sequence !
I'll be able to do this myself in a few days, but maybe for you it's not too complicated and I'll see the results a few days in advance... No problem for me if you want to stop now and not generate these new tables, because all this is really a lot of work and requires really a lot of time ! :smile:[/QUOTE] I think I have it working, but we'll have to see. It definitely knocks the size of the files down. Here is the entire base2primes for the new filtering (if I have it right: primes >10^4 that repeat within a sequence): [code] prime 10111 shows up 4 times ( 76:i10, 416:i61, 416:i81, 506:i6 ). prime 10613 shows up 3 times ( 148:i3, 148:i23, 516:i104 ). prime 10667 shows up 3 times ( 111:i19, 405:i47, 405:i59 ). prime 15121 shows up 5 times ( 220:i15, 270:i1, 309:i27, 540:i1, 540:i2 ). prime 37517 shows up 3 times ( 219:i28, 219:i56, 467:i8 ). [/code]Although base 2 shows only five primes meeting the criteria, base 3 has a few more at 1417. I'm running a full set for all the tables in hopes it will be done when I get up in the morning. When finished, I'll upload the files so you can see what you think. 
@EdH :
A lot of thanks for the base 2310. A lot of thanks for your new effort ! I look forward to the results ! For my part, I'm trying to reproduce your calculations so that the program execution time is reasonable... 
[SIZE=4][COLOR=Red]I think I am now finally able to formulate a general conjecture that encompasses the two little conjectures stated in posts #384 and #387.[/COLOR][/SIZE]
[U]General conjecture :[/U] [B]s(n) = sigma(n)n If p = s(3^i) is a prime, then we have :  s(3^(2i)) = m * p and s(s(3^(2i))) = m * r, where r is any integer. [/B][B] s(3^(2i * k)) = m * p * u and s(s(3^(2i * k))) = m * t, with u and t any integers but p and m which remain the same whatever k integer k>=1 for a given i.[/B] I hope I have made this conjecture clear in English ! For a better understanding, here are some numerical examples below (I looked up all the i<=500 such that p = s(3^i) is a prime number) : [CODE]i = 3 p = 13 m = 28 = 2^2 * 7 i = 7 p = 1093 m = 2188 = 2^2 * 547 i = 13 p = 797161 m = 1594324 = 2^2 * 398581 i = 71 p = 3754733257489862401973357979128773 m = 7509466514979724803946715958257548 = 2^2 * 853 * 2131 * 82219 * 3099719989 * 4052490063499 i = 103 p = 6957596529882152968992225251835887181478451547013 m = 13915193059764305937984450503671774362956903094028 = 2^2 * 619 * 3661040653 * 1535090713229126909942383374434289901 [/CODE]If we take for example i=71, we have : i=71 p=3754733257489862401973357979128773 m=7509466514979724803946715958257548=2^2*853*2131*82219*309919989*4052490063499 So, we can say that for all sequences that begin with 3^(2*71 * k) = 3^(142k), with k being an integer, we will find the factor p*m in the decomposition of the term at index 1 and we will find the factor m in the decomposition of the term at index 2. I'm not quite sure how to try to demonstrate this conjecture yet, I haven't spent any time on it. Either she's already known. Otherwise, it shouldn't be very difficult to prove it for someone who's used to this kind of problem... 
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[QUOTE=garambois;552595]@EdH :
A lot of thanks for the base 2310. A lot of thanks for your new effort ! I look forward to the results ! For my part, I'm trying to reproduce your calculations so that the program execution time is reasonable...[/QUOTE] Here is a set of (hopefully) all the primes >10^4 that repeat within a single sequence for all the current bases listed on the page. If the prime shows up in other sequences, those are listed as well, whether there are repetitions in the subsequent sequence(s) or not. Here is a familiar sample from the end of base 3: [code] prime 3099719989 shows up 2 times ( [B]142:i1, 142:i2[/B] ). prime 3661040653 shows up 2 times ( [B]206:i1, 206:i2[/B] ). prime 4052490063499 shows up 2 times ( [B]142:i1, 142:i2[/B] ). prime 1535090713229126909942383374434289901 shows up 2 times ( [B]206:i1, 206:i2[/B] ). [/code]Also in base 3: [code] prime 50077 shows up 4 times ( 74:i1145, 107:i1, [B]214:i1, 214:i2[/B] ). [/code]Here is a full list of the primes I found that show up in Indices 1 and 2 within a sequence: [code] base2primes:prime 15121 shows up 5 times ( 220:i15, 270:i1, 309:i27, [B]540:i1, 540:i2[/B] ). base3primes:prime 17761 shows up 10 times ( 14:i1550, 28:i2055, 68:i1359, 110:i2051, 140:i1820, [B]185:i1, 185:i2[/B], 204:i1259, 210:i18, 235:i56 ). base3primes:prime 50077 shows up 4 times ( 74:i1145, 107:i1, [B]214:i1, 214:i2[/B] ). base3primes:prime 82219 shows up 5 times ( 78:i861, [B]142:i1, 142:i2[/B], 150:i633, 204:i10450 ). base3primes:prime 398581 shows up 18 times ( [B]26:i1, 26:i2[/B], [B]52:i1, 52:i2[/B], [B]78:i1, 78:i2[/B], [B]104:i1, 104:i2[/B], [B]130:i1, 130:i2[/B], [B]156:i1, 156:i2[/B], [B]182:i1, 182:i2[/B], [B]208:i1, 208:i2[/B], [B]234:i1, 234:i2[/B] ). base3primes:prime 3099719989 shows up 2 times ( [B]142:i1, 142:i2[/B] ). base3primes:prime 3661040653 shows up 2 times ( [B]206:i1, 206:i2[/B] ). base3primes:prime 4052490063499 shows up 2 times ( [B]142:i1, 142:i2[/B] ). base3primes:prime 1535090713229126909942383374434289901 shows up 2 times ( [B]206:i1, 206:i2[/B] ). [/code] 
Thank you very much Ed for these precious new tables. I'm going to take a close look at them...
Right now, I have no idea what's going on with the prime number 50077. Either this number escapes conjecture, or the conjecture is still incomplete !!! 
[QUOTE=EdH;552556]. . .
In the meantime, I have decided to color in the transparent cells for base 2310 after all.[/QUOTE]They should all be colored in now. All the cofactors that remain have been ECMed to Aliqueit's fullest desires. 
Thank you very much.
There will be even more colors in the update in a few days ! :smile: 
[QUOTE=garambois;552909]Thank you very much.
There will be even more colors in the update in a few days ! :smile:[/QUOTE] I look forward to seeing all the colors! Are there more data points for consideration? A couple things interesting to me: 1. ATM, [URL="http://www.factordb.com/sequences.php?se=1&aq=2%5E544&action=range&fr=0&to=22"]2^544[/URL] has a "3" downdriver. 2. [URL="http://www.factordb.com/sequences.php?se=1&aq=2%5E543&action=range&fr=9&to=10"]2^543[/URL] shed 69 digits from index 9 to index 10 (15182 dd). 
[QUOTE=EdH;553029]I look forward to seeing all the colors!
Are there more data points for consideration? A couple things interesting to me: 1. ATM, [URL="http://www.factordb.com/sequences.php?se=1&aq=2%5E544&action=range&fr=0&to=22"]2^544[/URL] has a "3" downdriver. 2. [URL="http://www.factordb.com/sequences.php?se=1&aq=2%5E543&action=range&fr=9&to=10"]2^543[/URL] shed 69 digits from index 9 to index 10 (15182 dd).[/QUOTE] Unfortunately, I have not yet had the time to study this aspect of the sequences for the different bases. But I note these remarks for some time to come when I will start this kind of studies... I am working very intensively on prime numbers and their occurrence in the different sequences and it takes all my time. I finally have the right analysis programs. And everything corresponds perfectly to the results of EdH ! I think I will be able to present you with several new conjectures in the next few days, but before I do, I'd rather check them out. I will take some time to update the page very soon. So I have a question for Ed : @EdH : I understand you're working on the 2^i sequences with i>540. Should I add a line from 2^540 to 559 ? Otherwise, if possible the first priority would be to compute the first 4 indexes of the sequences 2^(36*k), 2^(60*k), 2^(70*k), with k integer and the first 3 indexes of the sequences 2^(70*k), 2^(72*k), 2^(90*k), with k integer and of course 36*k>540, 60*k>540, 70*k>540... (because when it's <540, we've already got them !) This has something to do with the conjectures I will be stating in a few days. For example, I observe that the prime number 5 is found in the decompositions of terms at indexes 1 to 4 of sequences that begin with 2^(36*k) or that the prime number 19 is found in the decompositions of terms at indexes 1 to 3 of sequences that begin with 2^(72*k). But I think that this must stop after a certain rank, so I would like to prove it... 
@JeanLuc: I will suspend my sequential march and look at your new request. These may be bordering on my capabilities quite quickly, though.
You have 70*k listed as both 4 and 3 indexes. Is one of these different? When you refer to indexes 1 through 4 are these the index number or the total number of factored lines so that you have the latest aliquot sum on index number 5? 
Sorry, for 70*k : 3 indexes and not 4.
Index 0 is 2^(70*k). index 1 is s(2^(70*k)) index 2 is s(s(2^(70*k))) index 3 is s(s(s(2^(70*k)))) So index 3 means 4 factored lines and index 4 means 5 factored lines. Thanks a lot Ed ! 
[QUOTE=richs;552064]Reserving 439^28 at i407.[/QUOTE]
439^28 is now at i603 (added almost 200 iterations) and a C121 level with a 2 * 3^3 guide, so I will drop this reservation. The remaining C105 term is well ecm'ed and is ready for siqs. Reserving 439^30 at i80. 
OK, page updated.
Many thanks to all... 
30^43 has surely merged. No way I could get that sequence to a C158 is such a short time.

[QUOTE=RichD;553343]30^43 has surely merged. No way I could get that sequence to a C158 is such a short time.[/QUOTE]
[code] 30^43:i2207 merges with 39060:i2 [/code] 
OK, thanks, I'll add this merger in the next update !

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[QUOTE=garambois;550094]The first array is finished.
You can see it as an attachment (.pdf version). It is difficult to draw definite conclusions because we don't have a lot of sequences that end for large bases after all. But what I was hoping for is not happening. [B]Sequences that start on integer powers seem to generally end with the same probability on the same prime numbers as all of the sequences.[/B] So there's no obvious conjecture to be made... yet. I will redo all this work by considering all the prime numbers that appear in all the terms of the sequences, as described above. I will also publish the final array here. If anyone has any questions or notices things that I wouldn't have seen when looking at this array, please feel free to express them here ![/QUOTE] As announced in post #337 cited above, here is the attached pdf which shows the occurrences of prime numbers <1000 for all the bases. Here, we consider globally all the terms of all the sequences for a base. It is rather the prime numbers 31 and 127 which are distinguished from the others. But this is understandable since they are the prime numbers of the drivers... I didn't fill in the column called "integers from 1 to 10^4". Downloading all the complete sequences on db would have been much too laborious ! [B]That said, it would certainly be extremely interesting to redo for all the integers all this prime number analysis work that we did for only the integer powers. Some very interesting things would certainly appear. I am talking about all the works, and not only those shown on this pdf and the pdf of post #337 (works also showing multiple apparitions of a prime number in a single sequence by indicating the indexes of appearance). Now that the programs are written, it would be easy to do the analyses for all the integers. The problem is the downloading of all the terms of all the sequences...[/B] 
I'm done with n=13, all sequences after 13^80 now are >120 digits with >110 digits composites passed ECM work.

All right, thank you very much.

@JeanLuc: I'm having difficulty with factordb for the higher orders of 2^[I]i[/I]. It won't download the elfs, which is how my scripts currently retrieve the composites to run. I think I've reached the digit level that I'll have to consider my upper bounds for now, anyway. I have done a few of the requested sequences, but not very far. Sorry about that!
What I have done includes [I]i[/I]=560, 576, 600, 612, 630 and 648. 
You may have exceeded one of the hourly limits for your userid and/or IP address.

[QUOTE=EdH;553560]
What I have done includes [I]i[/I]=560, 576, 600, 612, 630 and 648.[/QUOTE] Thanks a lot, Ed. It's huge already ! I couldn't have done this job ! All your calculations corroborate my conjectures, except for the calculation of the sequence that starts with 2^630 : it invalidates one of my conjectures !!! :smile: I'll present my observations very soon... 
[QUOTE=RichD;553561]You may have exceeded one of the hourly limits for your userid and/or IP address.[/QUOTE]
Nope. It's something to do with the operation of the db. Take a look at 2^600: [URL]http://www.factordb.com/sequences.php?se=1&aq=2%5E600&action=last20&fr=0&to=100[/URL] Then look at 2^684: [URL]http://www.factordb.com/sequences.php?se=1&aq=2%5E684&action=last20&fr=0&to=100[/URL] 2^600 has the two graph links and the "Download .elf file" link which downloads the .elf file, appropriately. They are centered on the line. 2^684 is missing the two graph links and the "Download .elf file" link is over on the left. Clicking acts like it's downloading, but no file shows up. wget acts the same: 2^600 d/ls fine, 2^684 is empty. I'd be interested in how it performs for you. 
[QUOTE=garambois;553563]Thanks a lot, Ed.
It's huge already ! I couldn't have done this job ! All your calculations corroborate my conjectures, except for the calculation of the sequence that starts with 2^630 : it invalidates one of my conjectures !!! :smile: I'll present my observations very soon...[/QUOTE] Bummer on 2^630! Blame it on me.:smile: Looking forward to the report(s). . . 
[QUOTE=EdH;553568]I'd be interested in how it performs for you.[/QUOTE]
Same spastic behavior. Once the beginning term is 200 digits or more that's when things go awry. 2^660 is fine but 2^662 (200 digits) produces the same results as your 2^684. 
[QUOTE=RichD;553572]Same spastic behavior. Once the beginning term is 200 digits or more that's when things go awry.
2^660 is fine but 2^662 (200 digits) produces the same results as your 2^684.[/QUOTE] Thanks RichD. If I do any more, I'll just have to manually work them. But, as I mentioned, I'm probably nearing my boundary. If ECM doesn't factor them at least into the 16x area, I'm not inclined to run GNFS at that level right now. 
[QUOTE=EdH;550299]Additionally, that [I]a[SUP]i[/SUP]+1 [/I]is a factor of[I][I] s(a[SUP](i*n)[/SUP]) [/I][/I]([I][I]n,[/I][/I] a positive even integer)[/QUOTE]
Please, Ed, is your proposal complete ? Because if we take, for example, a=10, n=4 and i=3, we have : s(10^(4*3)) = s(10^12) = 1499694822171 and 10^3+1 = 1001 And 1001 do not divide 1499694822171. 
I run some of the 3^i for i<=96. You should update them from time2time. No need to reserve them for me.

[QUOTE=garambois;553629]Please, Ed, is your proposal complete ?
Because if we take, for example, a=10, n=4 and i=3, we have : s(10^(4*3)) = s(10^12) = 1499694822171 and 10^3+1 = 1001 And 1001 do not divide 1499694822171.[/QUOTE] I was just tossing out a couple things that seemed of interest. I can't do more for a few days. If it's not valid, that's fine. I'll look at it more in a few days. 
@ yoyo :
Thanks a lot ! Just out of curiosity, are the calculations done by BOINC's yafu project ? @ EdH : OK. So I'm going to focus using my tables on the case of the odd numbers you mention in post #364. 
I don't remember being so hesitant in my work on aliquot sequences. I've been running my analysis algorithms for over a week now. Then finally, I decided several times to modify them to make the results more accessible or to make other phenomena appear.
And I observe a lot of conjectures. I observe so many that it is difficult to see which ones are the most interesting. I don't even know how to present all this to you clearly. I am preparing a huge post with a selection of some of the most representative conjectures. But I still need more time. My dream is to find a general conjecture, which was my original goal. For example, I dream of being able to relate occurrences of prime numbers for bases p and q (prime p and q) to occurrences of prime numbers for the base p*q. 
[QUOTE=garambois;553747]@ yoyo :
Thanks a lot ! Just out of curiosity, are the calculations done by BOINC's yafu project ? [/QUOTE] Yes they are handled by BOINC's yafu project. 
[QUOTE=yoyo;553750]Yes they are handled by BOINC's yafu project.[/QUOTE]
WAOUH, thank you !!! :tu: 
[QUOTE=EdH;553628]If I do any more, I'll just have to manually work them. But, as I mentioned, I'm probably nearing my boundary. If ECM doesn't factor them at least into the 16x area, I'm not inclined to run GNFS at that level right now.[/QUOTE]
I'm thinking if you can keep the first few terms locally then add the sequence to FDB once it drops below 200 digits. That requires a few extra steps but you could continue your analysis a bit further. 
[QUOTE=yoyo;553680]I run some of the 3^i for i<=96. You should update them from time2time. No need to reserve them for me.[/QUOTE]
Update: I take all 3^i which have composites < C140. 
[QUOTE=EdH;553726]I was just tossing out a couple things that seemed of interest. I can't do more for a few days. If it's not valid, that's fine. I'll look at it more in a few days.[/QUOTE]
I just reread post #379 : OK, a is not an even number ! I misunderstood ! I'm sorry ! The conjecture seems to be true but a must be prime. But I don't know how to prove it ! 
[QUOTE=yoyo;553830]Update: I take all 3^i which have composites < C140.[/QUOTE]
OK, thank you. For the next update, very soon, I'll reserve the orange cells of base 3 for you. Is that right ? For the name of the contributor, I put "YOY" ? 
Ok, you can put YOY on it.

You can remove my name from the colored cells (orange/green) in the n=30 table. I am done with them, so please release.

I do not deserve any credit for any of the green [I]n[/I]=21 sequences, as I did no work on those. I don't know if you (JeanLuc) did them or if they were done directly on FactorDB, but those shouldn't be attributed to me.

@Happy5244 :
Don't I get it ? I thought it was you ! Would you rather I replaced "HPP" with "A" for Anonymous ? Because I didn't do the sequence calculations ! Unless the person who did them comes forward... I hope I didn't make a mistake in the mention of the contributor for base 21. It's not voluntary if yes ! @RichD : OK, I'm releasing the orange cells. But the green ones will always keep our name, because they were finished by you... @yoyo : OK. 
[QUOTE=garambois;553995]@Happy5244 :
Don't I get it ? I thought it was you ! Would you rather I replaced "HPP" with "A" for Anonymous ? Because I didn't do the sequence calculations ! Unless the person who did them comes forward... I hope I didn't make a mistake in the mention of the contributor for base 21. It's not voluntary if yes ! [/QUOTE] My guess is that they were finished by FactorDB automatically when the sequences were pulled by your script. I think they should be marked Anonymous. 
OK, I'll note them "A".

Do you update the page by script or is it manually work?

I update the page by script.
But it still takes me more than an hour, because I have to check everything (because of bad handling sometimes), enter the reservations, I have to deal with the different bases separately, reread all the posts to take it into account ... And I'm missing a lot of time, like everybody else ;) 
[QUOTE=Happy5214;553989]I do not deserve any credit for any of the green [I]n[/I]=21 sequences, as I did no work on those. I don't know if you (JeanLuc) did them or if they were done directly on FactorDB, but those shouldn't be attributed to me.[/QUOTE]
[QUOTE=garambois;553995]@RichD : But the green ones will always keep our name, because they were finished by you...[/QUOTE] [QUOTE=garambois;554018]OK, I'll note them "A".[/QUOTE] I am seeing the same thing with base n=30. There was only a handful of exponents I had to advance to get the termination. Let's say any sequence with <= 25 terms (indexes) was created by FDB and should be credited to "A". 
OK, page updated.
Many thanks to all ! Base 14 complete. Base 510510 complete. I take base 30. @Happy5214 : At the next update, what name of contributor do you want me to put for the green cells of bases 24 and 210 ? @yoyo : Now we've got a white cell for Base 3 again ! 
Why is there no table for base 4, 8 and 9?

[QUOTE=garambois;554109]I take base 30.[/QUOTE]
I was still working on the white cells in base 30 but you can have it. 
I'll start table 770.

[QUOTE=yoyo;554113]Why is there no table for base 4, 8 and 9?[/QUOTE]
The sequences of bases 4 and 8 are the same as those of base 2. Indeed, 4^k=(2^2)^k=2^(2*k) and 8^k=(2^3)^k=2^(3*k) Same for base 9=3^2 with base 3. 
[QUOTE=RichD;553883]You can remove my name from the colored cells (orange/green) in the n=30 table. I am done with them, so please release.[/QUOTE]
Sorry about base 30, but when I read this message, I understood that you wanted to abandon it ! :confused: Ok for the new base 770, thank you very much ! I'll add it when you let me know you've done the preliminary work... 
[QUOTE=garambois;554109]@Happy5214 :
At the next update, what name of contributor do you want me to put for the green cells of bases 24 and 210 ? [/QUOTE] Those can be attributed to me except for the really small ones (< 3e6) that are in the main project. Even for the ones I didn't do locally, I got the ball rolling on FactorDB. The difference with [I]n[/I]=21 is that I didn't set up any of the initial data. You can also remove me from the sequences merging into main project sequences, since I don't plan to work on those and don't consider those reserved. 
OK, thank you.
All this will be done in the next update. 
Table 770
Table 770 is ready for insertion. Of course any “C” number that is working is subject to change at any time.
[CODE]i= Status   1 A 12 (green) 2 A 3 (green) 3 C146 (released) 4 A 4 (green) 5 C101 (working) 6 A 14 (green) 7 C99 (working) 8 A 21 (green) 9 C102 (working) 10 A 10 (green) 11 C87 (working) 12 RFD 30 (green) 13 RFD 1410 (green) 14 A 21 (green) 15 C100 (working) 16 RFD 29 (green) 17 C88 (working) 18 A 6 (green) 19 C100 (working) 20 A 23 (green) 21 C105 (working) 22 RFD 28 (green) 23 C98 (working) 24 RFD 38 (green) 25 C83 (working) 26 RFD 60 (green) 27 C85 (working) 28 RFD 36 (green) 29 C85 (working) 30 RFD 49 (green) 31 C93 (working) 32 RFD 43 (green) 33 C87 (working) 34 RFD 49 (green) 35 C90 (working) 36 RFD 71 (green) 37 C107 (working) 38 RFD 62 (green) 39 C115 (working) 40 RFD 38 (green) 41 C119 (working) 42 RFD 29 (green)[/CODE] 
Many thanks RichD !
Next update, in approximately one week... Until then, your calculations may still be evolving. 
[QUOTE=garambois;554109]
@yoyo : Now we've got a white cell for Base 3 again ![/QUOTE] What does this mean? My reserved sequences are now cycling through the system, between BOINC server and volunteers. 
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[SIZE=7][COLOR=Red]New conjectures[/COLOR][/SIZE]
In the first part of this long post, I propose some conjectures. Following the posts #384, #387 and #392, I add here some conjectures that I could formulate by observing the tables produced by the analysis algorithms of Edwin Hall and mine. I tried to classify these conjectures, but the classification is just intuitive and not based on real arguments.  The conjectures with a star (*): I think they are true and could probably be demonstrated by mathematicians working on number theory. Some of them are certainly already known or are even theorems. Sorry if some of these conjectures are trivial, I'm not necessarily able to realize it!  The conjectures with two stars (**): I think they are true. If they are indeed true, they must be very difficult to prove. But I may be wrong about some of them and further calculations should make it possible to invalidate them.  The conjectures without a star: they are more observations than conjectures. My intuition tells me that the continuation of the calculations would show that they are false. But you never know...  Bold conjecture is either very beautiful or spectacular! I have been unable to produce a general conjecture. The conjectures I propose here only concern occurrences of a prime number in a base. I cannot state all the conjectures, there are far too many. I have tried to select a representative sample. After the statement of the conjectures, in a second part, I propose an explanation which shows how I proceed from the observation of the tables, to state these conjectures. Finally, in the third part, I make some general remarks and ask some questions. [SIZE=6][U]I) Statement of new conjectures :[/U][/SIZE] In all the statements below, k is an integer. [SIZE=5][U]Some new conjectures for base 2:[/U][/SIZE] Note: Several of these conjectures motivated my request to Edwin Hall to push the calculations further for some exponents i=36*k, i=60*k, i=70*k, i=72*k, i=90*k. Conjecture (1)* : The prime number 3 appears in the decomposition of the terms of index 1 of all sequences that begin with the integers 2^(2*k). Conjecture (2)** : The prime number 3 appears in the decomposition of the terms of index 1, 2 of all sequences that begin with the integers 2^(4*k). Conjecture (3) : The prime number 3 appears in the decomposition of the terms of indexes 1 through 7 of all sequences that begin with the integers 2^(36*k). Conjecture (4) : The prime number 3 appears in the decomposition of the terms of indexes 1 through 18 of all sequences that begin with the integers 2^(126*k). Conjecture (5)* : The prime number 5 appears in the decomposition of the terms of index 1 of all sequences that begin with the integers 2^(4*k). Conjecture (6)** : The prime number 5 appears in the decomposition of the terms of index 1, 2 of all sequences that begin with the integers 2^(28*k), 2^(44*k), 2^(76*k), 2^(92*k), 2^(116*k). Conjecture (7) : The prime number 5 appears in the decomposition of the terms of indexes 1, 2, 3, 4 of all sequences that begin with the integers 2^(36*k). Conjecture (8) : The prime number 5 appears in the decomposition of the terms of indexes 1, 2, 3, 4 of all sequences that begin with the integers 2^(132*k). Conjecture (9)* : The prime number 7 appears in the decomposition of the terms of index 1 of all sequences that begin with the integers 2^(3*k). Conjecture (10)** : The prime number 7 appears in the decomposition of the terms of index 1, 2 of all sequences that begin with the integers 2^(12*k). Conjecture (11) : The prime number 7 appears in the decomposition of the terms of indexes 1, 2, 3, 4 of all sequences that begin with the integers 2^(60*k). Conjecture (12)* : The prime number 11 appears in the decomposition of the terms of index 1 of all sequences that begin with the integers 2^(10*k). Conjecture (13)** : The prime number 11 appears in the decomposition of the terms of index 1, 2 of all sequences that begin with the integers 2^(120*k), 2^(130*k). Conjecture (14) : The prime number 11 appears in the decomposition of the terms of indexes 1, 2, 3 of all sequences that begin with the integers 2^(70*k). Conjecture (15) : The prime number 13 appears in the decomposition of the terms of index 1 of all sequences that begin with the integers 2^(12*k). Conjecture (16)** : The prime number 13 appears in the decomposition of the terms of index 1, 2 of all sequences that begin with the integers 2^(60*k). Conjecture (17)* : The prime number 17 appears in the decomposition of the terms of index 1 of all sequences that begin with the integers 2^(8*k). Conjecture (18)** : The prime number 17 appears in the decomposition of the terms of index 1, 2 of all sequences that begin with the integers 2^(144*k). Conjecture (19) : The prime number 19 appears in the decomposition of the terms of indexes 1, 2, 3 of all sequences that begin with the integers 2^(72*k). [STRIKE]Conjecture (20) : The prime number 31 appears in the decomposition of the terms of indexes 1, 2, 3 of all sequences that begin with the integers 2^(90*k).[/STRIKE] Conjecture invalidated by Edwin Hall's calculations. [B]Conjecture (21)** : The prime number 79 appears in the decomposition of the terms of indexes 1, 2, 3 of all sequences that begin with the integers 2^(156*k).[/B] Conjecture (22)** : The prime number 2089 appears in the decomposition of the terms of indexes 1, 2 of all sequences that begin with the integers 2^(87*k). Conjecture (23)** : The prime number 4051 appears in the decomposition of the terms of indexes 1, 2 of all sequences that begin with the integers 2^(100*k). Conjecture (24) : The prime number 15121 appears in the decomposition of the terms of indexes 1, 2 of all sequences that begin with the integers 2^(540*k). [SIZE=5][U]Some new conjectures for base 3:[/U][/SIZE] Conjecture (25)*: The prime number 5 appears in the decomposition of the terms of index 1 of all sequences starting with the integers 3^(4*k). Conjecture (26)** : The prime number 5 appears in the decomposition of the terms of index 1, 2 of all sequences that begin with the integers 3^(4+8*k). Conjecture (27)* : The prime number 7 appears in the decomposition of the terms of index 1 of all sequences that begin with the integers 3^(6*k). Conjecture (28)* : The prime number 7 appears in the decomposition of the terms of many consecutive indexes of all sequences that begin with the integers 3^(6+12*k). Here is the observation that led to this conjecture: [CODE]prime 7 in sequence 3^6 at index i for i from 1 to 5 prime 7 in sequence 3^18 at index i for i from 1 to 50 prime 7 in sequence 3^30 at index i for i from 1 to 25 prime 7 in sequence 3^42 at index i for i from 1 to 86 prime 7 in sequence 3^54 at index i for i from 1 to 179 prime 7 in sequence 3^66 at index i for i from 1 to 39 prime 7 in sequence 3^78 at index i for i from 1 to 124 prime 7 in sequence 3^90 at index i for i from 1 to 171 prime 7 in sequence 3^102 at index i for i from 1 to 72 prime 7 in sequence 3^114 at index i for i from 1 to 45 prime 7 in sequence 3^126 at index i for i from 1 to 60 prime 7 in sequence 3^138 at index i for i from 1 to 230 prime 7 in sequence 3^150 at index i for i from 1 to 148 prime 7 in sequence 3^162 at index i for i from 1 to 228 prime 7 in sequence 3^174 at index i for i from 1 to 219 prime 7 in sequence 3^186 at index i for i from 1 to 9 prime 7 in sequence 3^198 at index i for i from 1 to 105 prime 7 in sequence 3^210 at index i for i from 1 to 194 prime 7 in sequence 3^222 at index i for i from 1 to 98 prime 7 in sequence 3^234 at index i for i from 1 to 87 prime 7 in sequence 3^246 at index i for i from 1 to 38[/CODE]But on reflection, this conjecture is not extraordinary. 7 is a prime number which is in the dcomposition of the 2^2*7 driver. It is therefore normal that it persists in so many consecutive terms. On the other hand, it should be shown here that s(3^(6+12*k)) has the driver 2^2*7 as a factor. Conjecture (29)* : The prime number 11 appears in the decomposition of the terms of index 1 of all sequences that begin with the integers 3^(5*k). Conjecture (30)* : The prime number 13 appears in the decomposition of index 1 terms in all sequences that begin with the integers 3^(3*k). Conjecture (31)** : The prime number 13 appears in the decomposition of the terms of index 1, 2 of all sequences that begin with the integers 3^(51*k). Conjecture (32)* : The prime number 17 appears in the decomposition of the terms of index 1 of all sequences that begin with the integers 3^(16*k). Conjecture (33)** : The prime number 17 appears in the decomposition of the terms of index 1, 2 of all sequences that begin with the integers 3^(48*k). Conjecture (34) (* if only index 1) : The prime number 19 appears in the decomposition of the terms of indexes 1, 2 of all sequences that begin with the integers 3^(18*k). Conjecture (35) (* if only index 1 and 2): The prime number 19 appears in the decomposition of the terms of indexes 1, 2, 3 of all sequences that begin with the integers 3^(36*k). Conjecture (36)* : The prime number 23 appears in the decomposition of the terms of index 1 of all sequences that begin with the integers 3^(11*k). Conjecture (37)** : The prime number 31 appears in the decomposition of the terms of index 1, 2 of all sequences that begin with the integers 3^(30*k). Conjecture (38)* : The prime number 37 appears in the decomposition of the terms of index 1 of all sequences that begin with the integers 3^(18*k). Conjecture (39)** : The prime number 37 appears in the decomposition of the terms of index 1, 2 of all sequences that begin with the integers 3^(36*k). [B]Conjecture (40)** : The prime number 79 appears in the decomposition of the terms of indexes 1, 2, 3 of all sequences that begin with the integers 3^(78*k). [/B] Conjecture (41)** : The prime number 547 appears in the decomposition of the terms of indexes 1, 2 of all sequences that begin with the integers 3^(14*k). Conjecture (42)**, already known conjecture, see previous posts : The prime number 398581 appears in the decomposition of the terms of indexes 1, 2, 3 of all sequences that begin with the integers 3^(26*k). [SIZE=5][U]Some new conjectures for base 5:[/U][/SIZE] Conjecture (43): The prime number 3 appears in the decomposition of the terms of index 1 of all sequences starting with the integers 5^(2*k). [B]Conjecture (44) : The prime number 3 appears in the decomposition of many consecutive indexes of all sequences that begin with the integers 5^(2+4*k). For example, 3 appears in the decomposition of the terms in indexes 1 through 786 of the sequence that begins with 5^58.[/B] Conjecture (45)* : The first number 5 never appears in the decomposition of the terms at index 1 of all sequences beginning with the integers 5^(k). Conjecture (46)* : The prime number 7 appears in the decomposition of terms at index 1 of all sequences that begin with the integers 5^(6*k). Conjecture (47)** : The prime number 7 appears in the decomposition of index 1 and index 2 terms of all sequences that begin with the integers 5^(12*k). Conjecture (48)* : The prime number 11 appears in the decomposition of the terms of index 1 of all sequences that begin with the integers 5^(5*k). Conjecture (49)** : The prime number 11 appears in the decomposition of the terms of index 1, 2 of all sequences that begin with the integers 5^(35*k). Conjecture (50)** : The prime number 11 appears in the decomposition of the terms of indexes 1, 2 of all sequences that begin with the integers 5^(40*k). Conjecture (51)** : The prime number 11 appears in the decomposition of the terms of indexes 1, 2 of all sequences that begin with the integers 5^(65*k). Conjecture (52)* : The prime number 13 appears in the decomposition of the terms of index 1 of all sequences that begin with the integers 5^(4*k). Conjecture (53)* : The prime number 17 appears in the decomposition of index 1 terms in all sequences that begin with the integers 5^(16*k). Conjecture (54)* : The prime number 19 appears in the decomposition of index 1 terms in all sequences that begin with the integers 5^(9*k). Conjecture (55)** : The prime number 19 appears in the decomposition of the terms of index 1, 2 of all sequences that begin with the integers 5^(18*k). Conjecture (56)* : The prime number 31 appears in the decomposition of the terms of index 1 of all sequences that begin with the integers 5^(3*k). Conjecture (57)** : The prime number 31 appears in the decomposition of the terms of index 1, 2 of all sequences that begin with the integers 5^(30*k). Conjecture (58) : The prime number 31 appears in the decomposition of the terms of many consecutive indexes of all sequences that begin with the integers 5^(48+96*k). Here is the observation that led to this conjecture: [CODE]prime 31 in sequence 5^48 at index i for i from 1 to 447 prime 31 in sequence 5^144 at index i for i from 1 to 32[/CODE]The same remark can be made here as for the conjecture (28). And it should be shown here that s(5^(48+96*k)) has the driver 2^4*31 as a factor. Conjecture (59)* : The prime number 71 appears in the decomposition of the terms of index 1 of all sequences that begin with the integers 5^(5*k). Conjecture (60)** : The prime number 71 appears in the decomposition of the terms of index 1, 2 of all sequences that begin with the integers 5^(45*k). Conjecture (61)* : The prime number 521 appears in the decomposition of the terms of index 1 of all sequences that begin with the integers 5^(10*k). Conjecture (62)** : The prime number 521 appears in the decomposition of the terms of index 1, 2 of all sequences that begin with the integers 5^(50*k). [SIZE=5][U]Some new conjectures for base 6:[/U][/SIZE] Conjecture (63): The prime number 3 appears in the decomposition of the terms of index 1 of all sequences that start with the integers 6^(1+2*k). Conjecture (64)* : The prime number 5 appears in the decomposition of the terms of index 1 of all sequences that begin with the integers 6^(2*k). Conjecture (65)* : The prime number 7 appears in the decomposition of index 1 terms of all sequences that begin with the integers 6^(6*k). Conjecture (66)* : The prime number 11 appears in the decomposition of index 1 terms of all sequences that begin with the integers 6^(10*k) and 6^(2+10*k). Conjecture (67)* : The prime number 13 appears in the decomposition of the terms of index 1 of all sequences that begin with the integers 6^(12*k). Conjecture (68)* : The prime number 19 appears in the decomposition of index 1 terms of all sequences that begin with the integers 6^(18*k) and 6^(10 + 18*k). Conjecture (69)* : The prime number 23 appears in the decomposition of the terms of index 1 of all sequences that begin with the integers 6^(11*k). Conjecture (70)* : The prime number 29 appears in the decomposition of index 1 terms in all sequences that begin with the integers 6^(28*k). Conjecture (71)* : The prime number 31 appears in the decomposition of index 1 terms of all sequences that begin with the integers 6^(30*k), 6^(11+30*k) and 6^(17 + 30*k). Conjecture (72)** : The prime number 37 appears in the decomposition of the terms of indexes 1, 2 of all sequences that begin with the integers 6^(36*k) Conjecture (73)* : The prime number 37 appears in the decomposition of the terms of index 1 of all sequences starting with the integers 6^(14+36*k). Conjecture (74)* : The prime number 59 appears in the decomposition of the terms of index 1 of all sequences that begin with the integers 6^(58*k), 6^(8+58*k), 6^(35+58*k) and 6^(53+58*k). Conjecture (75)* : The prime number 61 appears in the decomposition of the terms of index 1 of all sequences beginning with the integers 6^(60*k), 6^(44+60*k) and 6^(55+60*k). Conjecture (76)* : The prime number 71 appears in the decomposition of the terms of index 1 of all sequences that begin with the integers 6^(70*k), 6^(11+70*k), 6^(32+70*k), 6^(35+70*k), 6^(46+70*k) and 6^(67+70*k). Conjecture (77)* : The prime number 601 appears in the decomposition of the terms of index 1 of all sequences beginning with the integers 6^(75*k). [SIZE=5][U]Some new conjectures for Base 7:[/U] [/SIZE] Conjecture (78): The prime number 3 appears in the decomposition of the terms of index 1 of all sequences starting with the integers 7^(3*k). Conjecture (79) : The prime number 3 appears in the decomposition of the terms from index 1 to 10 for all sequences that begin with the integers 5^(6+12*k) and 5^(21*k). Conjecture (80)* : The prime number 5 appears in the decomposition of the terms of index 1 for all sequences that begin with the integers 7^(4*k). Conjecture (81)* : The prime number 7 appears in the decomposition of index 1 terms in all sequences that begin with the integers 7^(3*k). Conjecture (82)* : The prime number 7 never appears in index 1 of all sequences that begin with the integers 7^(k). Conjecture (83)* : The prime number 11 appears in the decomposition of the terms in index 1 of all sequences that begin with the integers 7^(10*k). Conjecture (84)** : The prime number 13 appears in the decomposition of the terms of index 1, 2 of all sequences that begin with the integers 7^(12*k). [B]Conjecture (85) : The prime number 13 appears in the decomposition of the terms of many indexes of all sequences that begin with the integers 7^(72*k). 27 consecutive indexes for 7^72 and 9 consecutive indexes for 7^144.[/B] Conjecture (86)* : The prime number 17 appears in the decomposition of the terms of index 1 of all sequences that begin with the integers 7^(16*k). Conjecture (87)** : The prime number 17 appears in the decomposition of the terms of index 1, 2 of all sequences that begin with the integers 7^(32*k). Conjecture (88)* : The prime number 19 appears in the decomposition of the terms of index 1 of all sequences that begin with the integers 7^(3*k). Conjecture (89)** : The prime number 19 appears in the decomposition of the terms of index 1, 2 of all sequences that begin with the integers 7^(9*k). Conjecture (90)* : The prime number 31 appears in the decomposition of the terms of index 1 of all sequences that begin with the integers 7^(15*k). Conjecture (91)** : The prime number 67 appears in the decomposition of the terms of index 1, 2 of all sequences that begin with the integers 7^(66*k). Conjecture (92)* : The prime number 419 appears in the decomposition of the terms of index 1 of all sequences that begin with the integers 7^(19*k). [B]Conjecture (93)** : The prime number 419 appears in the decomposition of the terms of index 1, 2 of all sequences that begin with the integers 7^(38*k). Checked up to k=23.[/B] [SIZE=5][U]Some new conjectures for base 10:[/U][/SIZE] Conjecture (94): The prime number 3 appears in the decomposition of the terms of index 1 of all sequences starting with the integers 10^(2*k). Conjecture (95)* : The prime number 5 appears in the decomposition of the terms of index 1 of all sequences that begin with the integers 10^(3+4*k). Conjecture (96)* : The prime number 13 appears in the decomposition of the terms of index 1 of all sequences that begin with the integers 10^(12*k). Conjecture (97)** : The prime number 13 appears in the decomposition of the terms of index 1, 2 of all sequences that begin with the integers 10^(2+12*k). Conjecture (98)** : The prime number 19 appears in the decomposition of the terms of indexes 1, 2 of all sequences that begin with the integers 10^(18*k). Conjecture (99)* : The prime number 61 appears in the decomposition of the terms of index 1 of all sequences that begin with the integers 10^(52+60*k) and 10^(54+60*k). Conjecture (100)** : The prime number 61 appears in the decomposition of the terms of index 1, 2 of all sequences that begin with the integers 10^(60*k). [SIZE=5][U]Some new conjectures for base 11 :[/U][/SIZE] Conjecture (101)*: The prime number 7 appears in the decomposition of the terms of index 1 of all sequences that begin with the integers 11^(3*k). Conjecture (102)** : The prime number 7 appears in the decomposition of the terms of indexes 1, 2, 3 of all sequences that begin with the integers 11^(6*k). Conjecture (103)* : The prime number 11 never appears in the decomposition of the terms of index 1 of all sequences that begin with the integers 11^(k). [B]Conjecture (104)** : The prime number 13 appears in the decomposition of the terms of index 1, 2 of all sequences that begin with the integers 11^(k).[/B] Conjecture (105)* : The prime number 19 appears in the decomposition of the terms of index 1 of all sequences that begin with the integers 11^(3*k). Conjecture (106)** : The prime number 19 appears in the decomposition of the terms of index 1, 2 of all sequences that begin with the integers 11^(6*k). [B]Conjecture (107)** : The product of prime 19*79*547 appears in the decomposition of the terms of indexes 1, 2 of all sequences that begin with the integers 11^(39*k). REMARKABLE, checked up to k=12. See prime 79 bases 2 and 3.[/B] [SIZE=5][U]Some new conjectures for base 12:[/U][/SIZE] Conjecture (108): The prime number 3 never appears in the decomposition of the terms of index 1 of all sequences that start with the integers 12^(k). Conjecture (109)* : The prime number 17 appears in the decomposition of index 1 terms in all sequences that begin with the integers 12^(16*k) and 12^(6+16*k). It is difficult to notice for base 12, other behaviors different from the bases already presented so far. [SIZE=5][U]Some new conjectures for base 13:[/U] [/SIZE] Conjecture (110)* : The prime number 3 appears in the decomposition of the terms of index 1 of all sequences that start with the integers 13^(3*k). [B]Conjecture (111) : The prime number 3 appears in the decomposition of the terms of indexes 1 through 6 of all sequences that begin with the integers 13^(6*k).[/B] Conjecture (112)* : The prime number 5 appears in the decomposition of the terms in index 1 of all sequences that begin with the integers 13^(4*k). Conjecture (113) : The prime number 5 appears in the decomposition of the terms of many consecutive indexes of all sequences that begin with the integers 13^(8+16*k). Conjecture (114)* : The prime number 7 appears in the decomposition of the terms of index 1 of all sequences that begin with the integers 13^(2*k). Conjecture (115) : The prime number 7 appears in the decomposition of the terms of many consecutive indexes of all sequences that begin with the integers 13^(4+8*k). Conjecture (116)* : The prime number 13 never appears in the decomposition of the terms of index 1 of all sequences that begin with the integers 13^(k). Conjecture (117)* : The prime number 19 appears in the decomposition of index 1 terms in all sequences that begin with the integers 13^(18*k). Conjecture (118)** : The prime number 19 appears in the decomposition of the terms of index 1, 2 of all sequences that begin with the integers 13^(36*k). Conjecture (119)* : The prime number 29 appears in the decomposition of the terms of index 1 of all sequences that begin with the integers 13^(14*k). Conjecture (120)** : The prime number 29 appears in the decomposition of the terms of index 1, 2 of all sequences that begin with the integers 13^(42*k). Conjecture (121)* : The prime number 61 appears in the decomposition of the terms of index 1 of all sequences that begin with the integers 13^(3*k). Conjecture (122)** : The prime number 61 appears in the decomposition of the terms of index 1, 2 of all sequences that begin with the integers 13^(21*k). [SIZE=5][U]Some new conjectures for base 14:[/U] [/SIZE] [B]Conjecture (123)** : The prime number 3 appears in the decomposition of the terms of indexes 1 to 4 of all sequences starting with the integers 14^(6*k).[/B] Conjecture (124)* : The prime number 5 appears in the decomposition of the terms of index 1 of all sequences that begin with the integers 14^(4*k). Conjecture (125)** : The prime number 5 appears in the decomposition of the terms of indexes 1 to 4 of all sequences that begin with the integers 14^(1+4*k). It is difficult to notice for base 14, other behaviors different from the bases already presented so far... [SIZE=5][U]Some new conjectures for base 15 :[/U][/SIZE] Conjecture (126)* : The prime number 7 appears in the decomposition of the terms of index 1 of all sequences which start with the integers 15^(2*k), except for the 15^(8+12*k). [SIZE=5][U]Some new conjectures for base 17 :[/U][/SIZE] Conjecture (127)** : The prime number 3 appears in the decomposition of the terms of indexes 1, 2 of all sequences that begin with the integers 17^(2*k). Conjecture (128)** : The prime number 5 appears in the decomposition of the terms of indexes 1, 2 of all sequences that begin with the integers 17^(4*k). Conjecture (129)** : The prime number 7 appears in the decomposition of the terms of indexes 1, 2 of all sequences that begin with the integers 17^(6*k). Conjecture (130)** : The prime number 19 appears in the decomposition of the terms of index 1 of all sequences that begin with the integers 17^(9*k). Conjecture (131)** : The prime number 19 appears in the decomposition of the terms of index 1, 2 of all sequences that begin with the integers 17^(36*k). Conjecture (132)** : The prime number 229 appears in the decomposition of the terms of index 1 of all sequences that begin with the integers 17^(19*k). Conjecture (133)** : The prime number 229 appears in the decomposition of the terms of index 1, 2 of all sequences that begin with the integers 17^(38*k). [U][SIZE=5]Other bases : See note 5 below.[/SIZE][/U] [SIZE=6]II) How do I proceed to state these conjectures from the tables?[/SIZE] To illustrate the difficulty of stating these conjectures, I suggest you look at the attached files named "base_x_mat" and "base_x_matcons". First we had to find the way to display the data in the easiest way possible. "base_x_mat" shows the occurrences of all the prime numbers appearing in the whole database, specifying in which sequence they appear and at which indexes. "base_x_matcons" is an extract from "base_x_mat" and shows only those prime numbers that appear at consecutive indexes starting from 1 (or more rarely 0 or 2) for a given sequence. Attached are the file pairs of base 2, 3, 5, 6, 7, 10, 11, 12, 13, 14, 15, 17 (the last two files base_15_mat and base_17_mat are missing for reasons of size allowed on the forum). For example, when we see this extract from the "base_3_matcons" file : [CODE]prime 79 in sequence 3^78 at index 1 2 3 prime 79 in sequence 3^156 at index 1 2 3 prime 79 in sequence 3^234 at index 1 2 3[/CODE]The conjecture is immediately deduced from this (40). But it is much more difficult to deduce conjectures (5), (6), (7) and (8) from this excerpt from the "base_2_matcons" file. [CODE]prime 5 in sequence 2^4 at index 1 prime 5 in sequence 2^8 at index 1 prime 5 in sequence 2^12 at index 1 prime 5 in sequence 2^16 at index 1 prime 5 in sequence 2^20 at index 1 prime 5 in sequence 2^24 at index 1 prime 5 in sequence 2^28 at index 1 2 prime 5 in sequence 2^32 at index 1 prime 5 in sequence 2^36 at index 1 2 3 4 5 6 prime 5 in sequence 2^40 at index 1 prime 5 in sequence 2^44 at index 1 2 prime 5 in sequence 2^48 at index 1 prime 5 in sequence 2^52 at index 1 prime 5 in sequence 2^56 at index 1 2 3 prime 5 in sequence 2^60 at index 1 prime 5 in sequence 2^64 at index 1 prime 5 in sequence 2^68 at index 1 prime 5 in sequence 2^72 at index 1 2 3 4 5 prime 5 in sequence 2^76 at index 1 2 3 prime 5 in sequence 2^80 at index 1 prime 5 in sequence 2^84 at index 1 2 prime 5 in sequence 2^88 at index 1 2 prime 5 in sequence 2^92 at index 1 2 prime 5 in sequence 2^96 at index 1 prime 5 in sequence 2^100 at index 1 prime 5 in sequence 2^104 at index 1 prime 5 in sequence 2^108 at index 1 2 3 4 5 6 7 8 prime 5 in sequence 2^112 at index 1 2 prime 5 in sequence 2^116 at index 1 2 prime 5 in sequence 2^120 at index 1 prime 5 in sequence 2^124 at index 1 prime 5 in sequence 2^128 at index 1 prime 5 in sequence 2^132 at index 1 2 3 4 5 6 prime 5 in sequence 2^136 at index 1 2 3 prime 5 in sequence 2^140 at index 1 2 3 4 prime 5 in sequence 2^144 at index 1 2 3 4 5 6 7 8 9 10 11 12 13 prime 5 in sequence 2^148 at index 1 2 prime 5 in sequence 2^152 at index 1 2 3 4 5 6 7 8 9 10 prime 5 in sequence 2^156 at index 1 2 3 4 prime 5 in sequence 2^160 at index 1 prime 5 in sequence 2^164 at index 1 2 3 4 5 prime 5 in sequence 2^168 at index 1 2 3 4 5 6 prime 5 in sequence 2^172 at index 1 2 3 4 5 prime 5 in sequence 2^176 at index 1 2 3 4 5 6 7 prime 5 in sequence 2^180 at index 1 2 3 4 5 prime 5 in sequence 2^184 at index 1 2 3 prime 5 in sequence 2^188 at index 1 2 prime 5 in sequence 2^192 at index 1 prime 5 in sequence 2^196 at index 1 2 prime 5 in sequence 2^200 at index 1 prime 5 in sequence 2^204 at index 1 2 3 4 prime 5 in sequence 2^208 at index 1 prime 5 in sequence 2^212 at index 1 2 3 4 prime 5 in sequence 2^216 at index 1 2 3 4 5 6 prime 5 in sequence 2^220 at index 1 2 prime 5 in sequence 2^224 at index 1 2 3 4 5 prime 5 in sequence 2^228 at index 1 2 3 4 5 prime 5 in sequence 2^232 at index 1 2 prime 5 in sequence 2^236 at index 1 2 3 prime 5 in sequence 2^240 at index 1 prime 5 in sequence 2^244 at index 1 2 3 4 5 prime 5 in sequence 2^248 at index 1 prime 5 in sequence 2^252 at index 1 2 3 4 5 prime 5 in sequence 2^256 at index 1 prime 5 in sequence 2^260 at index 1 prime 5 in sequence 2^264 at index 1 2 3 4 prime 5 in sequence 2^268 at index 1 2 prime 5 in sequence 2^272 at index 1 2 3 4 5 6 prime 5 in sequence 2^276 at index 1 2 3 prime 5 in sequence 2^280 at index 1 2 prime 5 in sequence 2^284 at index 1 2 3 prime 5 in sequence 2^288 at index 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 prime 5 in sequence 2^292 at index 1 2 prime 5 in sequence 2^296 at index 1 2 3 4 5 prime 5 in sequence 2^300 at index 1 prime 5 in sequence 2^304 at index 1 2 3 4 5 prime 5 in sequence 2^308 at index 1 2 3 4 5 6 7 prime 5 in sequence 2^312 at index 1 2 3 4 5 prime 5 in sequence 2^316 at index 1 2 prime 5 in sequence 2^320 at index 1 prime 5 in sequence 2^324 at index 1 2 3 4 prime 5 in sequence 2^328 at index 1 2 3 4 prime 5 in sequence 2^332 at index 1 2 3 prime 5 in sequence 2^336 at index 1 2 3 4 5 6 7 prime 5 in sequence 2^340 at index 1 prime 5 in sequence 2^344 at index 1 2 prime 5 in sequence 2^348 at index 1 2 3 4 5 6 prime 5 in sequence 2^352 at index 1 2 3 4 5 prime 5 in sequence 2^356 at index 1 2 prime 5 in sequence 2^360 at index 1 2 3 4 5 6 7 8 9 10 prime 5 in sequence 2^364 at index 1 2 prime 5 in sequence 2^368 at index 1 2 3 4 5 6 7 8 9 prime 5 in sequence 2^372 at index 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 prime 5 in sequence 2^376 at index 1 2 3 4 5 6 7 8 9 10 11 12 13 14 prime 5 in sequence 2^380 at index 1 2 3 4 prime 5 in sequence 2^384 at index 1 2 3 4 prime 5 in sequence 2^388 at index 1 2 3 4 5 6 prime 5 in sequence 2^392 at index 1 2 3 prime 5 in sequence 2^396 at index 1 2 3 4 5 6 7 8 9 prime 5 in sequence 2^400 at index 1 prime 5 in sequence 2^404 at index 1 2 3 4 5 6 7 8 prime 5 in sequence 2^408 at index 1 2 prime 5 in sequence 2^412 at index 1 2 3 4 5 6 7 8 9 10 prime 5 in sequence 2^416 at index 1 2 prime 5 in sequence 2^420 at index 1 2 3 prime 5 in sequence 2^424 at index 1 2 3 prime 5 in sequence 2^428 at index 1 2 3 prime 5 in sequence 2^432 at index 1 2 3 4 5 6 7 8 prime 5 in sequence 2^436 at index 1 2 prime 5 in sequence 2^440 at index 1 2 3 4 5 6 7 prime 5 in sequence 2^444 at index 1 2 3 4 5 6 7 prime 5 in sequence 2^448 at index 1 2 3 4 5 prime 5 in sequence 2^452 at index 1 2 prime 5 in sequence 2^456 at index 1 2 3 prime 5 in sequence 2^460 at index 1 2 prime 5 in sequence 2^464 at index 1 2 3 prime 5 in sequence 2^468 at index 1 2 3 4 5 6 7 8 9 prime 5 in sequence 2^472 at index 1 2 3 prime 5 in sequence 2^476 at index 1 2 3 prime 5 in sequence 2^480 at index 1 prime 5 in sequence 2^484 at index 1 2 prime 5 in sequence 2^488 at index 1 2 prime 5 in sequence 2^492 at index 1 2 3 4 5 6 7 8 prime 5 in sequence 2^496 at index 1 2 3 4 5 prime 5 in sequence 2^500 at index 1 prime 5 in sequence 2^504 at index 1 2 3 4 5 prime 5 in sequence 2^508 at index 1 2 3 prime 5 in sequence 2^512 at index 1 prime 5 in sequence 2^516 at index 1 2 3 prime 5 in sequence 2^520 at index 1 prime 5 in sequence 2^524 at index 1 2 3 4 prime 5 in sequence 2^528 at index 1 2 3 4 5 6 7 8 prime 5 in sequence 2^532 at index 1 2 3 4 5 prime 5 in sequence 2^536 at index 1 2 3 4 prime 5 in sequence 2^540 at index 1 2 3 4 5 6 7[/CODE]I failed in my attempt to write an algorithm that can automatically find guesses. I spot them by looking at the files, but after a few hours, my eyes start to sting. Hence question 1 below. [SIZE=6]III) Remarks and questions[/SIZE] [SIZE=4][U]Remark 1:[/U][/SIZE] My long term goal would have been to find a relationship between the occurrences of prime numbers for the bases p and q (prime p and q) with the occurrences of prime numbers in the compound base p*q. For example: is there a relationship between the occurrences of prime numbers for bases 3 and 5 and between the occurrences of prime numbers for the base 3*5=15? [SIZE=4][U]Remark 2:[/U][/SIZE] All this work is very complicated. It is difficult to judge which conjectures are interesting. The interest in the end, would be to find occurrences for large prime numbers, to accelerate much the factorizations of the terms of the sequences. [SIZE=4][U]Remark 3:[/U][/SIZE] Unfortunately, I did not succeed in exploiting the lines of the tables whose indexes are not consecutive for a prime number in a given sequence. Is there anything to notice with these lines? [SIZE=4][U]Remark 4:[/U][/SIZE] I think Ed's statements (post #364 and post #379) are true. But I don't know how to prove them. [SIZE=4][U]Remark 5:[/U][/SIZE] I've only presented in this post the conjectures up to base 17. Writing conjectures is very constraining and takes a lot of time. [B]But if someone has interest in the other bases, just ask me. I can join in another post all the tables and continue to write the conjectures properly.[/B] [SIZE=4][U]Remark 6:[/U][/SIZE][B] I certainly forgot some conjectures. I may not have seen the most important ones. A general conjecture may be obvious! The whole mode can try to find conjectures from the tables! Now it's up to you to try your luck...[/B] [SIZE=4][U]Remark 7:[/U][/SIZE] The work is far from over. I have other ideas to test ! [SIZE=4][U]Question 1:[/U][/SIZE] How to write a program that automates conjectures from tables (see example above with the prime number 5 in base 2 and the file "base_2_matcons") ? [SIZE=4][U]Question 2 :[/U][/SIZE] How to prove this type of result: If n=3^(6+12*k), then s(n) is divisible by 2^2*7 ? This may not be very difficult for a professional mathematician, but I don't know how to do it ! [SIZE=4][U]Question 3 :[/U][/SIZE] How to prove (or invalidate) otherwise than by calculation, this kind of result (conjectures often noted **): If n=3^(78*k), then s(n), s(s(n)) and s(s(s(n))) are all three divisible by 79? This kind of proof must be much, much more difficult than the one we are talking about in question 2! 
Wow! I will be studying all this for quite a while.:smile:
[QUOTE=garambois;554224]. . . [SIZE=4][U]Question 1:[/U][/SIZE] How to write a program that automates conjectures from tables (see example above with the prime number 5 in base 2 and the file "base_2_matcons") ? . . .[/QUOTE] If I'm reading this question correctly, you can use grep. Here is an example: [code] $ cat base_2_matcons  grep "prime 5 "  grep "x 1 2 3 4 5" [/code]Result: [code] prime 5 in sequence 2^36 at index 1 2 3 4 5 6 prime 5 in sequence 2^72 at index 1 2 3 4 5 prime 5 in sequence 2^108 at index 1 2 3 4 5 6 7 8 prime 5 in sequence 2^132 at index 1 2 3 4 5 6 prime 5 in sequence 2^144 at index 1 2 3 4 5 6 7 8 9 10 11 12 13 prime 5 in sequence 2^152 at index 1 2 3 4 5 6 7 8 9 10 prime 5 in sequence 2^164 at index 1 2 3 4 5 prime 5 in sequence 2^168 at index 1 2 3 4 5 6 prime 5 in sequence 2^172 at index 1 2 3 4 5 prime 5 in sequence 2^176 at index 1 2 3 4 5 6 7 prime 5 in sequence 2^180 at index 1 2 3 4 5 prime 5 in sequence 2^216 at index 1 2 3 4 5 6 prime 5 in sequence 2^224 at index 1 2 3 4 5 prime 5 in sequence 2^228 at index 1 2 3 4 5 prime 5 in sequence 2^244 at index 1 2 3 4 5 prime 5 in sequence 2^252 at index 1 2 3 4 5 prime 5 in sequence 2^272 at index 1 2 3 4 5 6 prime 5 in sequence 2^288 at index 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 prime 5 in sequence 2^296 at index 1 2 3 4 5 prime 5 in sequence 2^304 at index 1 2 3 4 5 prime 5 in sequence 2^308 at index 1 2 3 4 5 6 7 prime 5 in sequence 2^312 at index 1 2 3 4 5 prime 5 in sequence 2^336 at index 1 2 3 4 5 6 7 prime 5 in sequence 2^348 at index 1 2 3 4 5 6 prime 5 in sequence 2^352 at index 1 2 3 4 5 prime 5 in sequence 2^360 at index 1 2 3 4 5 6 7 8 9 10 prime 5 in sequence 2^368 at index 1 2 3 4 5 6 7 8 9 prime 5 in sequence 2^372 at index 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 prime 5 in sequence 2^376 at index 1 2 3 4 5 6 7 8 9 10 11 12 13 14 prime 5 in sequence 2^388 at index 1 2 3 4 5 6 prime 5 in sequence 2^396 at index 1 2 3 4 5 6 7 8 9 prime 5 in sequence 2^404 at index 1 2 3 4 5 6 7 8 prime 5 in sequence 2^412 at index 1 2 3 4 5 6 7 8 9 10 prime 5 in sequence 2^432 at index 1 2 3 4 5 6 7 8 prime 5 in sequence 2^440 at index 1 2 3 4 5 6 7 prime 5 in sequence 2^444 at index 1 2 3 4 5 6 7 prime 5 in sequence 2^448 at index 1 2 3 4 5 prime 5 in sequence 2^468 at index 1 2 3 4 5 6 7 8 9 prime 5 in sequence 2^492 at index 1 2 3 4 5 6 7 8 prime 5 in sequence 2^496 at index 1 2 3 4 5 prime 5 in sequence 2^504 at index 1 2 3 4 5 prime 5 in sequence 2^528 at index 1 2 3 4 5 6 7 8 prime 5 in sequence 2^532 at index 1 2 3 4 5 prime 5 in sequence 2^540 at index 1 2 3 4 5 6 7 [/code]Using base_2_mat shows the following: [code] $ cat base_2_mat  grep "prime 5 "  grep "x 1 2 3 4 5" prime 5 in sequence 2^36 at index 1 2 3 4 5 6 8 9 10 12 13 prime 5 in sequence 2^72 at index 1 2 3 4 5 9 14 15 22 23 26 27 prime 5 in sequence 2^108 at index 1 2 3 4 5 6 7 8 17 18 19 prime 5 in sequence 2^132 at index 1 2 3 4 5 6 16 27 34 35 40 41 prime 5 in sequence 2^144 at index 1 2 3 4 5 6 7 8 9 10 11 12 13 21 prime 5 in sequence 2^152 at index 1 2 3 4 5 6 7 8 9 10 20 21 24 28 29 30 31 32 36 37 38 39 41 42 47 48 49 50 52 53 prime 5 in sequence 2^164 at index 1 2 3 4 5 15 16 26 29 30 40 prime 5 in sequence 2^168 at index 1 2 3 4 5 6 14 15 16 22 25 26 prime 5 in sequence 2^172 at index 1 2 3 4 5 20 31 32 33 34 35 36 37 66 71 prime 5 in sequence 2^176 at index 1 2 3 4 5 6 7 16 17 18 24 prime 5 in sequence 2^180 at index 1 2 3 4 5 7 8 9 10 11 12 13 14 17 18 29 33 34 35 40 prime 5 in sequence 2^216 at index 1 2 3 4 5 6 9 10 12 19 20 29 30 31 32 prime 5 in sequence 2^224 at index 1 2 3 4 5 21 22 23 24 25 26 27 28 29 prime 5 in sequence 2^228 at index 1 2 3 4 5 30 31 35 36 37 38 39 40 41 42 43 44 45 48 53 54 57 61 67 68 69 74 prime 5 in sequence 2^244 at index 1 2 3 4 5 18 19 20 24 25 26 prime 5 in sequence 2^252 at index 1 2 3 4 5 8 31 32 33 42 54 prime 5 in sequence 2^272 at index 1 2 3 4 5 6 10 11 12 13 14 15 20 21 24 25 26 prime 5 in sequence 2^288 at index 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 29 37 39 40 43 49 prime 5 in sequence 2^296 at index 1 2 3 4 5 7 prime 5 in sequence 2^304 at index 1 2 3 4 5 prime 5 in sequence 2^308 at index 1 2 3 4 5 6 7 21 22 23 82 83 97 98 99 100 prime 5 in sequence 2^312 at index 1 2 3 4 5 14 15 16 17 18 19 26 27 28 30 31 38 39 40 41 50 51 52 57 59 60 61 62 66 67 68 79 82 83 84 prime 5 in sequence 2^336 at index 1 2 3 4 5 6 7 9 30 31 32 prime 5 in sequence 2^348 at index 1 2 3 4 5 6 9 10 14 15 16 17 24 28 prime 5 in sequence 2^352 at index 1 2 3 4 5 26 27 28 29 44 prime 5 in sequence 2^360 at index 1 2 3 4 5 6 7 8 9 10 33 34 35 36 37 38 39 40 prime 5 in sequence 2^368 at index 1 2 3 4 5 6 7 8 9 46 47 63 64 82 83 84 85 87 88 99 100 101 103 105 106 prime 5 in sequence 2^372 at index 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 prime 5 in sequence 2^376 at index 1 2 3 4 5 6 7 8 9 10 11 12 13 14 19 33 34 35 39 40 41 42 43 44 54 55 56 62 65 prime 5 in sequence 2^388 at index 1 2 3 4 5 6 12 36 37 38 39 64 72 73 75 79 80 87 prime 5 in sequence 2^396 at index 1 2 3 4 5 6 7 8 9 24 25 26 27 30 31 32 38 39 49 50 51 52 53 55 56 57 58 59 75 prime 5 in sequence 2^404 at index 1 2 3 4 5 6 7 8 12 13 14 15 16 17 prime 5 in sequence 2^412 at index 1 2 3 4 5 6 7 8 9 10 51 52 53 55 56 57 58 59 74 76 78 79 83 84 89 90 92 112 113 114 129 130 131 132 133 134 prime 5 in sequence 2^432 at index 1 2 3 4 5 6 7 8 13 14 15 16 17 18 19 30 31 32 33 34 35 36 38 39 prime 5 in sequence 2^440 at index 1 2 3 4 5 6 7 22 23 26 35 36 38 44 47 prime 5 in sequence 2^444 at index 1 2 3 4 5 6 7 27 28 29 30 31 32 33 35 36 37 38 39 45 prime 5 in sequence 2^448 at index 1 2 3 4 5 32 33 34 35 36 37 prime 5 in sequence 2^468 at index 1 2 3 4 5 6 7 8 9 37 38 41 48 74 78 prime 5 in sequence 2^492 at index 1 2 3 4 5 6 7 8 21 38 62 63 64 81 82 83 84 85 86 114 115 116 prime 5 in sequence 2^496 at index 1 2 3 4 5 prime 5 in sequence 2^504 at index 1 2 3 4 5 18 21 22 prime 5 in sequence 2^528 at index 1 2 3 4 5 6 7 8 15 23 24 25 26 27 28 39 40 41 46 47 48 49 50 51 54 55 56 57 64 65 66 67 68 69 70 71 72 78 79 80 81 82 83 85 86 87 88 89 90 92 93 102 103 104 105 106 107 116 117 118 119 140 141 147 169 173 177 179 180 183 184 prime 5 in sequence 2^532 at index 1 2 3 4 5 16 17 23 24 25 45 46 49 prime 5 in sequence 2^540 at index 1 2 3 4 5 6 7 12 13 14 15 16 17 18 19 20 21 22 23 31 32 33 65 70 71 73 75 76 77 78 79 80 92 93 94 100 103 104 [/code] 
@garambois: Which lets say 200 next sequences do you need?

[QUOTE=garambois;554224][COLOR=Red]New conjectures[/COLOR]
[SIZE=4][U]Question 2 :[/U][/SIZE] How to prove this type of result: If n=3^(6+12*k), then s(n) is divisible by 2^2*7 ? This may not be very difficult for a professional mathematician, but I don't know how to do it ![/QUOTE] For p prime, s(p^k) is (p^k1)/(p1). So S(3^(6+12*k)) is (3^(6+12*k)1)/2, which is (3^61)/2 * (3^(12*k)+3^(12k6)+3^(12k12)...+1) So s(n) divided by (3^61)/2 = 364 = 2^2 *7 * 13. As 3^6 is 1 (mod 364), each term of (3^(12*k)+3^(12k6)+3^(12k12)...+1) is (1 mod 364). Therefore 3^(12*k)+3^(12k6)+3^(12k12)...+1 is 2k+1 (mod 364) So if 2k+1 is not divided by either 7 or 13, we get a factor of 2^2*7*13 in s(n) If 2k+1 is divided by 7, such as in k=3, we get factor of 2^2*7^2*13 or more. Similarly, If 2k+1 is divided by 13, such as in k=6, we get factor of 2^2*7*13^2 or more. [QUOTE=garambois;554224][COLOR=Red]New conjectures[/COLOR] [U]Question 3 :[/U] How to prove (or invalidate) otherwise than by calculation, this kind of result (conjectures often noted **): If n=3^(78*k), then s(n), s(s(n)) and s(s(s(n))) are all three divisible by 79? This kind of proof must be much, much more difficult than the one we are talking about in question 2![/QUOTE] In general, if p and q (both prime) divide n, and p divide q+1, then p divide s(n). In case of n=3^(78*k), s(n) is (3^(78*k)1/)2, which is divided by (3^781)/2 = 2^2 · 7 · 13^2 · 79 · 157 · 313 · 2887 · 6553 · 7333 · 10141 · 398581 · 797161 Notice that 79 divide 157+1, and 157 divide 313+1, so 79 and 157 also divide s(s(n)), so 79 also divide s(s(s(n))). 
[QUOTE=yoyo;554252]@garambois: Which lets say 200 next sequences do you need?[/QUOTE]
@[B]garambois[/B]: I can release the 18 open sequences from table 770 to [B]yoyo[/B] and initialize tables 1155 and 385 or some other table(s) of your interest. Then repeat, release and initialize. You may have some resources coming your way... 
[QUOTE=warachwe;554258]
In general, if p and q (both prime) divide n, and p divide q+1, then p divide s(n). [/QUOTE] I forgot to mention that q^2 also must not divide n, or else p may not divide s(n). [QUOTE=warachwe;554258]In case of n=3^(78*k), s(n) is (3^(78*k)1/)2, which is divided by (3^781)/2 = 2^2 · 7 · 13^2 · 79 · 157 · 313 · 2887 · 6553 · 7333 · 10141 · 398581 · 797161 Notice that 79 divide 157+1, and 157 divide 313+1, so 79 and 157 also divide s(s(n)), so 79 also divide s(s(s(n))).[/QUOTE] So in this case, for s(s(s(n))) may not be divided by 79 if, for example, s(s(n)) got a factor of 157^2, but that's unlikely. 
[QUOTE=yoyo;554220]What does this mean?
[/QUOTE] That means the sequence's coming down and we might be lucky if it ends ! [QUOTE=yoyo;554220] My reserved sequences are now cycling through the system, between BOINC server and volunteers.[/QUOTE] OK, fantastic ! :smile: 
[QUOTE=EdH;554244]
If I'm reading this question correctly, you can use grep. Here is an example: [code] $ cat base_2_matcons  grep "prime 5 "  grep "x 1 2 3 4 5" [/code][/QUOTE] I didn't know the instruction "grep", I'm ashamed ! Thank you for this new tool for me ! The "grep" instruction doesn't give exactly the result I expect but I have no doubt that it will be very useful to me... 
@EdH :
Please, is it possible to modify question 3 of post #447 like this:[CODE]Question 3 : How to prove (or invalidate) otherwise than by calculation, this kind of result (conjectures often noted **): If n=3^(78*k), then s(n), s(n) and s(n)) are all three divisible by 79? This kind of proof must be much, much more difficult than the one we are talking about in question 2! [/CODE]should be :[CODE]Question 3 : How to prove (or invalidate) otherwise than by calculation, this kind of result (conjectures often noted **): If n=3^(78*k), then s(n), s(s(n)) and s(s(s(n))) are all three divisible by 79? This kind of proof must be much, much more difficult than the one we are talking about in question 2! [/CODE] And please, is it possible to modify also in the same way the statement of question 3 in the second "quote" of post #450. Machine translation played tricks on me ! But warachwe magnificently anticipated and corrected himself !!! 
[QUOTE=warachwe;554258]For p prime, s(p^k) is (p^k1)/(p1). So S(3^(6+12*k)) is (3^(6+12*k)1)/2, which is (3^61)/2 * (3^(12*k)+3^(12k6)+3^(12k12)...+1)
So s(n) divided by (3^61)/2 = 364 = 2^2 *7 * 13. As 3^6 is 1 (mod 364), each term of (3^(12*k)+3^(12k6)+3^(12k12)...+1) is (1 mod 364). Therefore 3^(12*k)+3^(12k6)+3^(12k12)...+1 is 2k+1 (mod 364) So if 2k+1 is not divided by either 7 or 13, we get a factor of 2^2*7*13 in s(n) If 2k+1 is divided by 7, such as in k=3, we get factor of 2^2*7^2*13 or more. Similarly, If 2k+1 is divided by 13, such as in k=6, we get factor of 2^2*7*13^2 or more. [/QUOTE] Okay, thank you so much for your help! I suspected this kind of result could be demonstrated. This implies that a large part of my conjectures (almost all those with a star) are in fact proven. [QUOTE=warachwe;554258] In general, if p and q (both prime) divide n, and p divide q+1, then p divide s(n). In case of n=3^(78*k), s(n) is (3^(78*k)1/)2, which is divided by (3^781)/2 = 2^2 · 7 · 13^2 · 79 · 157 · 313 · 2887 · 6553 · 7333 · 10141 · 398581 · 797161 Notice that 79 divide 157+1, and 157 divide 313+1, so 79 and 157 also divide s(s(n)), so 79 also divide s(s(s(n))).[/QUOTE] There was an error in the wording of question 3 of post #447, but you anticipated the real question I wanted to ask... Here again, thank you very much for your help ! I am really very surprised ! I thought that my conjectures noted with two stars implied mechanisms impossible to understand. Your explanations, as well as those of post #452 show that the conjectures noted with two stars are certainly not true, because if we continue the calculations long enough, sooner or later, for a rather large k, for example in the case of 3^(78*k), we will find the factor 157^2 in s(s(n)). These explanations significantly change the order of priorities for the calculations for the rest of the project. Above all, these explanations exempt us from continuing to look for this kind of conjectures. 
[SIZE=4]What are we going to do now ?[/SIZE]
If I take into account the events of the last few days, the study of the occurrences of prime numbers in all the terms of a sequence no longer seems to be a priority. I will tackle this problem in the light of the explanations in post #452. It therefore seems that it will be more useful to focus on the prime numbers that end sequences. We have to try to find at last some conjectures not yet known !!! However, the sequences that we are fairly certain that they end with a prime number are those that begin with numbers n^i when n and i have the same parity. If n and i don't have the same parity, some sequences also end but they are much rarer, but this makes them very interesting. So I propose : 1) To try to continue to advance (or even better, to end) sequences that have orange boxes for the different bases. 2) To start new bases and especially bases that are prime numbers:  Base 18  Base 19  Base 20  Base 23  Base 29  Base 31  Base 79 (seems to play a special role !) 3) To try to extend for each base the number of cells n^i with n and i having the same parity. I think to extend to the next update some bases in the following way : [CODE]for base 3 i from 252 to 335 for base 5 i from 172 to 229 for base 6 i from 155 to 206 for base 7 i from 142 to 189 for base 10 i from 121 to 160 for base 11 i from 116 to 153 for base 12 i from 113 to 148 for base 13 i from 108 to 143 for base 14 i from 105 to 140 for base 15 i from 104 to 139 for base 17 i from 98 to 131[/CODE]Then, [QUOTE=RichD;554336]@[B]garambois[/B]: I can release the 18 open sequences from table 770 to [B]yoyo[/B] and initialize tables 1155 and 385 or some other table(s) of your interest. Then repeat, release and initialize. You may have some resources coming your way...[/QUOTE] @RichD or others : Is it possible for you (or others who wish to do so) to do the preliminary work for either of the bases presented in point 2) ? [QUOTE=yoyo;554252]@garambois: Which lets say 200 next sequences do you need?[/QUOTE] @ yoyo : Is it possible for you at first to continue working on the orange cells on base 3? (point 1) above). And if you are willing to devote more computing resources to this project, you can add the orange cells from another base, for example base 5 (especially the cells that are stopped at 120 or 121 digits). Just let me know so that I can note the reservations on the page. With regard to point 3), I know that this request is not popular because it is very resourceintensive. But I think it is an important point. Let's wait until we see the update to get a good visualization of the work to be done. Everyone will then be able to see if they want to start according to their possibilities. 
I ran everything until the remaining composite is greater than C139.
Currently running up to 3^250 and 5^250 Seems that 5^185 5^179 have finished so far. If I need more for my hungry volunteers I would take base 6 and 7. 
[QUOTE=garambois;554359][2) To start new bases and especially bases that are prime numbers:
 Base 18  Base 19  Base 20  Base 23  Base 29  Base 31  Base 79 (seems to play a special role !)[/QUOTE] I will start preliminary work on Base 18 & Base 19. Work will (slowly) continue on Base 770 in the background. 
[QUOTE=garambois;554351]@EdH :
Please, is it possible to modify question 3 of post #447 like this: . . . [/QUOTE]Done. [QUOTE=garambois;554351] . . . And please, is it possible to modify also in the same way the statement of question 3 in the second "quote" of post #450. . . .[/QUOTE] @warachwe: May I edit garambois' quote in your post 450 to match the current version in post #447? 
Base 18
Base 18 table is ready for insertion. I took n to 96. Everything n <= 64 terminates (green). n > 64 has remaining composites near C80 or above and will need more work. They are all odd so they most likely will terminate. I will add them to the work queue which includes Base 770.

While we wait for the next update, to get a feel for the work relating to proposition 3, I will do the preliminary work for table 79.

Oh no! I've got i and n reversed in my above posts. I'll try to keep them straight going forward.
Table n=19 has been initialized to i=96 but there is still work to do. Many small remaining composites which I compete with the elves. Don't want to duplicate work. I'll let you know when I get it to a somewhat stable state. BTW, all the i=odd sequences terminate or mostly will terminate. But you already knew that. :smile: In the meantime I can start building Table n=23 next. Well, I mean initialize in FDB... 
OK, thanks to all for your big, big help !!!
I'm going to do a big update tomorrow to get a clearer view. [QUOTE=yoyo;554363] Currently running up to 3^250 and 5^250[/QUOTE] Are you sure of your exponent "250", because 5^250 has 175 digits ? 
[QUOTE=EdH;554365]Done.
@warachwe: May I edit garambois' quote in your post 450 to match the current version in post #447?[/QUOTE] Sure :smile: [QUOTE=garambois;554354]Your explanations, as well as those of post #452 show that the conjectures noted with two stars are certainly not true, because if we continue the calculations long enough, sooner or later, for a rather large k, for example in the case of 3^(78*k), we will find the factor 157^2 in s(s(n)).[/QUOTE] Those conjectures may still be true, because we can still get a factor of 79 from other primes than 157. For example, from conjecture 2), normally 5 is what provide a factor of 3 for s(n). But when k=5 (3^20) and we get a factor 5^2, we also get factor of 11 and 41, both of which also provide a factor of 3 for s(n). In this case I think conjecture 2) is true, but I can't prove it. P.S. Using [url=https://brilliant.org/wiki/liftingtheexponent/#:~:text=The%20%22lifting%20the%20exponent%22%20(,(x%2Cy)%20such%20that]LTE[/url] we find that 157^2 will divide s(3^(78*k)) if and only if 157 divide k (though we maybe have more than 157^2). 
From post #364
[QUOTE=EdH;550299]Does this apparent observation fit in with a similar theorem? For all [I]a[SUP]i[/SUP][/I] ([I]a[/I], [I]i[/I] positive integers ≥ 1) [I]s[/I]([I]a[SUP]i[/SUP][/I]) is a factor of [I]s[/I]([I]a[SUP](i*n)[/SUP][/I]) (for all positive n) [/QUOTE] for p prime, s(p[SUP]i[/SUP]) = (p[SUP]i[/SUP]1)/(p1) and s(p[SUP](i*n)[/SUP]) = (p[SUP](i*n)[/SUP]1)/(p1). Since (p[SUP]i[/SUP]1) is a factor of (p[SUP](i*n)[/SUP]1), s(p[SUP]i[/SUP]) is a factor of s(p[SUP](i*n)[/SUP]) for all positive integer n. for odd a in general, this is not true. For example [code] s(15[SUP](3)[/SUP]) = 3 · 5 · 191 s(15[SUP](3*2)[/SUP]) = 2 · 7 · 751 · 947 s(15[SUP](3*3)[/SUP]) = 3 · 76091 · 147353 [/code] 
Table n=23 has been initialized to i=88. Tables 19 & 23 can be instead in the web page with the understanding there is going to be many moving targets. FDB workers will factor small composites and I will slowly tackle the larger ones (up to C120).
In summary: Table n=18  all i <=77 terminate. Eventually all cells will most likely be green. Table n=19  It appears all i=odd sequences has or will terminate. Table n=23  It appears all i=odd sequences has or will terminate. 
[QUOTE=garambois;554399]Are you sure of your exponent "250", because 5^250 has 175 digits ?[/QUOTE]
Yes, my system skipped it fast, because the composite is larger than C139. I now go up to 3^335. Nothing in base 6 was reserved. So I took all ub to 6^206. 
I take also up to 7^189.

OK, page updated.
Thank you all very much for your work ! Thank you for pointing out possible handling errors. Base 18 added. Base 19 added. Base 23 added. Bases 3, 5, 6, 7, 10, 11, 12, 13, 14, 15 and 17 have been extended. @RichD : Can you please confirm that the attributions are good for green cells ? All cells in base 18 are green, because if n=18^i, s(n) becomes odd for all i. Indeed, n here is always even and we know that there is a change of parity if n is a perfect square or twice a perfect square. If i is even, then n is a perfect square, this is obvious. If i is odd, we pose i=2*k+1 and thus 18^(2*k+1)=(2*3^2)^(2*k+1)=2^(2*k+1)*3^(2*(2*k+1))=2*2^(2*k)*3^(2*(2*k+1)) which is thus a double of a perfect square. Thus, for the bases 18=2*3^2, 50=2*5^2, 72=2*6^6, 96=2*7^2... all squares will be green ! 
[QUOTE=warachwe;554404]
Those conjectures may still be true, because we can still get a factor of 79 from other primes than 157. For example, from conjecture 2), normally 5 is what provide a factor of 3 for s(n). But when k=5 (3^20) and we get a factor 5^2, we also get factor of 11 and 41, both of which also provide a factor of 3 for s(n). In this case I think conjecture 2) is true, but I can't prove it. P.S. Using [URL="https://brilliant.org/wiki/liftingtheexponent/#:~:text=The%20%22lifting%20the%20exponent%22%20(,(x%2Cy)%20such%20that"]LTE[/URL] we find that 157^2 will divide s(3^(78*k)) if and only if 157 divide k (though we maybe have more than 157^2).[/QUOTE] I'm beginning to understand the mechanism, thanks again to you for your explanations ! Your reasoning seems to be able to establish what we can guess with our calculations. Seeing this, I tell myself that it is no longer worth continuing to formulate such conjectures, like those of post #447. Therefore, we can focus on other phenomena. But, do you also think that it is no longer worthwhile to formulate conjectures like those of post #447, or do you think that this kind of conjecture can still be of interest ? On the other hand, don't hesitate to let us know if you have an idea to test based on our data that you might have had ! 
[QUOTE=yoyo;554363]
If I need more for my hungry volunteers I would take base 6 and 7.[/QUOTE] The yafu@home stats have been skyrocketing for the last few days ! :smile: 
[QUOTE=garambois;554494]@RichD : Can you please confirm that the attributions are good for green cells ?[/QUOTE]
I would say if a sequence has less than 25 terms, it was handled by FDB workers. I simply hit the refresh button and watch the sequence drop. No calculations by me. On the other hand, if a sequence starts say above C75, then I need to run the calculations to get it down to where the workers can finish it off. That being said, for Table n=18, "A" should be added to "i=" 610, 12,14, 16, 1822, 2426, 3031, 33, 36, 38, 40, 4243, 49, 57. Likewise for n=19 and n=23. 
Factordb itself factors everything below 70 digits. At present I'm factoring the range from 70 to 80 digits. So I've probably contributed to several sequences.
But there's been a lot of what looks like junk added to factordb over the past year of so. Numbers like 24681280*46##+251 obviously won't be part of a sequence. Does anyone know where they are coming from? Is there a way to check if a given number is from a sequence (eg clicking [c]More information[/c])? Chris 
factordb does factor <71 digit composites (and larger via helper elves), but Aliquot sequences are only advanced by bumping them. The bump may advance a few lines at a time, but if it stalls, it stays there even after the stalled composite is factored, until another bump. I have been doing all my sequence runs off line with Aliqueit to lessen the elves' load (having served as an elf myself).:smile:
Alas, I have found no way to tie a number to a particular sequence within the db. I suppose that's not feasible because of how many sequences would be listed for smaller numbers. 
Conjecture 2 is bugging me. It is fairly easy to show that the 5 maintains the 3 unless the sequence begins 2^(4*5^(2n1)*k).
If the sequence begins with 2^(4*5^n*k) then we know 2^201 = 3*5^2*11*31*41 is a divisor of s(2^(4*5^n*k)). This means we have to account for 11 and 41. In the same way as 5, 11 maintains the 3 unless the sequence begins 2^(4*5^(2n1)*11^(2m1)*k). Accounting for 41 gives a requirement of beginning with 2^(4*5^(2n1)*11^(2m1)*41^(2o1)*k). s(2^(4*5^(2n1)*11^(2m1)*41^(2o1)*k)) is always divisible by 2^90201 which contains yet more primes that preserve the 3. Expanding the lower bound like this is easy but I can't quite think how to turn it into a proof. Maybe there is something based on proving there will always be more and more prime factors that are 2 mod 3. 
[QUOTE=EdH;554527]... but Aliquot sequences are only advanced by bumping them.[/QUOTE]
That's what I meant by a refresh  a bump. :smile: 
[QUOTE=RichD;554540]That's what I meant by a refresh  a bump. :smile:[/QUOTE]
I understood that. I was replying to Chris.:smile: 
I was clarifying my original post for Chris. :smile:

[QUOTE=garambois;554497]
But, do you also think that it is no longer worthwhile to formulate conjectures like those of post #447, or do you think that this kind of conjecture can still be of interest ? [/QUOTE] Personally, I think these conjectures are interesting in a way that the prove seem somewhat doable, but I agree that we should look more into other phenomena. [QUOTE=henryzz;554530]Conjecture 2 is bugging me. It is fairly easy to show that the 5 maintains the 3 unless the sequence begins 2^(4*5^(2n1)*k). If the sequence begins with 2^(4*5^n*k) then we know 2^201 = 3*5^2*11*31*41 is a divisor of s(2^(4*5^n*k)). This means we have to account for 11 and 41. In the same way as 5, 11 maintains the 3 unless the sequence begins 2^(4*5^(2n1)*11^(2m1)*k). Accounting for 41 gives a requirement of beginning with 2^(4*5^(2n1)*11^(2m1)*41^(2o1)*k). s(2^(4*5^(2n1)*11^(2m1)*41^(2o1)*k)) is always divisible by 2^90201 which contains yet more primes that preserve the 3. Expanding the lower bound like this is easy but I can't quite think how to turn it into a proof. Maybe there is something based on proving there will always be more and more prime factors that are 2 mod 3.[/QUOTE] I think I found a way. S(2[SUP]4k[/SUP]) = 2[SUP]4k[/SUP]1=(2[SUP]2k[/SUP]1)(2[SUP]2k[/SUP]+1). 2[SUP]2k[/SUP]+1 is 2 (mod 3) , so there exist a prime p such that p is 2 (mod 3) and p[SUP]2m1 [/SUP] divide 2[SUP]2k[/SUP]+1 ,but p[SUP]2m [/SUP] not divide 2[SUP]2k[/SUP]+1 gcd(2[SUP]2k[/SUP]1,2[SUP]2k[/SUP]+1)=gcd(2,2[SUP]2k[/SUP]+1)=1, so no other factor of p from 2[SUP]2k[/SUP]1. Hence p[SUP]2m1 [/SUP] preserved 3 for s(2[SUP]4k[/SUP]), so 3 divide s(s(2[SUP]4k[/SUP])) 
[QUOTE=warachwe;554577]Personally, I think these conjectures are interesting in a way that the prove seem somewhat doable, but I agree that we should look more into other phenomena.
I think I found a way. S(2[SUP]4k[/SUP]) = 2[SUP]4k[/SUP]1=(2[SUP]2k[/SUP]1)(2[SUP]2k[/SUP]+1). 2[SUP]2k[/SUP]+1 is 2 (mod 3) , so there exist a prime p such that p is 2 (mod 3) and p[SUP]2m1 [/SUP] divide 2[SUP]2k[/SUP]+1 ,but p[SUP]2m [/SUP] not divide 2[SUP]2k[/SUP]+1 gcd(2[SUP]2k[/SUP]1,2[SUP]2k[/SUP]+1)=gcd(2,2[SUP]2k[/SUP]+1)=1, so no other factor of p from 2[SUP]2k[/SUP]1. Hence p[SUP]2m1 [/SUP] preserved 3 for s(2[SUP]4k[/SUP]), so 3 divide s(s(2[SUP]4k[/SUP]))[/QUOTE] Brilliant. Now to think about conjecture 3 
Table 18
All cells in n=18 should now be green.
I'll start preliminary work on Table n=20. 
The preliminary work on base 79 has been finished through [I]i[/I]=79.
Here is a list of all the prime terminations encountered: [code] 7 (1) 11 (1) 13 (1) 19 (1) 23 (1) 37 (1) 41 (3) 43 (1) 79 (1) 109 (1) 277 (1) 373 (1) 647 (1) 2053 (1) 3209 (1) 56417 (1) 274201 (1) 1404653 (1) 39449441 (1) 3726897833 (1) 15149890507 (1) 7003404108271 (1) 1750258119649999 (1) 50030448965430187 (1) 582160055705443697 (1) 737287429537747247 (1) 11218513019153099281 (1) 51295583920298014697 (1) 126184888328246734874859217 (1) 281050352710245559603460861942264287 (1) 612334801072206651528322236763533211 (1) 144410932495341043816499009968224193057 (1) 5215871181653976380322743907608410822343857033 (1) 10098441202754878623864731377586611856943613601 (1) 58715257934974315365589544756228858137685070659 (1) 4514080124904825140989752683500717658279017890401 (1) 144247534946592562812896661356728761240274391235398539509968323519 (1) 1543397657597549030942319129765250282733366278065161881086980020577 (1) 21210775475881297038103544006601558250238621656859058445205548774369 (1) 969294053696923120396927330592354921021401157551137054187134211303939583373802685159736925954726601223772976624910063809731121765969 (1)[/code]As shown, only the prime 41 terminated more than one sequence. 
@warachwe :
Thank you for your informed response (post #480) ! @ Everyone : For the moment, I'm not making any more conjectures like the ones in post #447. I'm trying to spot other different things, especially concerning the occurrences of prime numbers that end sequences. And also, for n=2^i, for n=18^i, for n=m^i, with m and i in the same parity, I know that s(n) is odd and therefore the sequence that starts with n most likely ends. But, I don't understand why all these sequences [U]without exception[/U] end. I'm looking forward to the first OpenEnd sequence for n=2^i, for n=18^i, for n=m^i, with m and i in the same parity ! But we'll have to wait until the amount of data we have increases. But with the computing power available to us at the moment, I estimate that in one week, we'll make more progress than in several years at the rate of a few months ago ! That said, I have no idea what "enough data" means. Is it going to take several months, a year, or more ? @warachwe & henryzz : If I understood correctly, the conjecture (2) is demonstrated ! Wow! Thanks to you ! Regarding conjecture (3), I didn't put a star. Because I doubt that the prime number 3 is kept indefinitely during the first 7 iterations for all the sequences starting with 2^(36*k). The calculations have not really been pushed beyond k=15 yet... But you never know ! @RichD and EdH : Thanks a lot for your help ! :smile: 
[QUOTE=garambois;554688]. . .
@ Everyone : For the moment, I'm not making any more conjectures like the ones in post #447. I'm trying to spot other different things, especially concerning the occurrences of prime numbers that end sequences. And also, for n=2^i, for n=18^i, for n=m^i, with m and i in the same parity, I know that s(n) is odd and therefore the sequence that starts with n most likely ends. But, I don't understand why all these sequences [U]without exception[/U] end. I'm looking forward to the first OpenEnd sequence for n=2^i, for n=18^i, for n=m^i, with m and i in the same parity ! But we'll have to wait until the amount of data we have increases. But with the computing power available to us at the moment, I estimate that in one week, we'll make more progress than in several years at the rate of a few months ago ! That said, I have no idea what "enough data" means. Is it going to take several months, a year, or more ? . . . :smile:[/QUOTE] 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. I'm working on some scripts/programs that will list all the terminating primes for a base, such as I supplied for base 79. I will be completing these for all the bases in the table. I have the scripts written such that when it runs, if the local .elf does not show a termination, the script downloads a fresh .elf. That ensures that any new terminations should be caught. Once I have a set of all the terminations for all the tables, we should be able to crossmatch as well. Let me know where you would prefer I work and anything you might like me to try to harvest with my scripts/programs. 
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