20200628, 16:53  #320 
Oct 2011
5×53 Posts 
@ EdH :
Okay, I'll make all the changes in the next update, very soon... @ Happy 5214 : Yes, your last idea was also suggested to me by Karsten Bonath. So the colors like now: All: x [Green: Done] [Red: Open] [Blue: Cycle] And then: All: x [Green: Done] [Red: Open] [Yellow: merged] [Blue: Cycle] And it also seems to me that this solution is the clearest and, as Karsten says, we don't need another explanation to understand. Do you think that if we proceed in this way, you will be able to do the work that motivated you to make this request ? 
20200629, 08:46  #321 
"Alexander"
Nov 2008
The Alamo City
349 Posts 
Will the merged sequences be taken out of the open count? I think doublecounting the merged sequences in both totals would be confusing. Overall, this plan works.
In other news, I've finished work on my n=3 reservations and am releasing those. 
20200629, 17:53  #322 
Oct 2011
109_{16} Posts 
OK, page updated.
A lot of thanks to all for your help. And once again, a big thank you to Karsten Bonath who sent me the modifications of the scripts. I'm not comfortable in this area and if I had to make the changes myself, it would most likely take me several hours or even days. And there's no guarantee that my changes would be done successfully... I have already once caused a disaster by wanting to touch the scripts myself ! Please, don't forget to check for updates that concern you personally and report any errors to me. 
20200706, 17:12  #323 
"Ed Hall"
Dec 2009
Adirondack Mtns
CB6_{16} Posts 
I have been playing with some programs/scripts to harvest some statistics from the sequences within the tables. Here is a sample of some individual statistics from the base 8128 table:
Code:
Stats for 8128^1: The complete (current) last line is: 2 . 8128 = 2^6 * 127 Total abundant terms: 0 Total deficient terms: 0 Most sequencial abundant terms: 0 Most sequencial deficient terms: 0 Parity changes even to odd: 0 Parity changes odd to even: 0 The first term is perfect. . . . Stats for 8128^30: The complete (current) last line is: 32 . 2460259041035273743806735759303370898982607278814358187347453 = 2460259041035273743806735759303370898982607278814358187347453 Total abundant terms: 1 Total deficient terms: 31 Most sequencial abundant terms: 1 Most sequencial deficient terms: 31 Parity changes even to odd: 1 Parity changes odd to even: 0 The first term is abundant. I chose 8128 for this example because it has only 30 sequences and is therefore somewhat smaller than others. If there is interest in these statistics, I can provide files for the other bases. These may be spread out over time. Additionally, if there are other things that may be of interest, I can try to add those items to my stats. This particular document shows individual statistics, but an aggregate for an entire table can also be harvested from it, or from a subsequent document with more specifics. An example would be capturing the last prime for all the prime terminated sequences for a count/comparison of common terminations within a table. I can probably find a way to show any perfect endings (as well as current beginnings) also, but I'm still struggling with how to catch cycles of greater than 1. Let me know of any more things of interest and I'll see if I can figure out how to add them. 
20200707, 08:15  #324 
Oct 2011
5·53 Posts 
EdH,
I'm very interested in your work ! Could you please provide me with the files for all the bases ? This is exactly the kind of work I started to do to examine the data. Your data is very complete compared to my work. I only added one more piece of data: the number of relative maximums in the "Huge graph" of the sequence. But this number should almost correspond to the number "Parity changes even to odd" with some exceptions. It can be different because a sequence can increase with odd terms and decrease with even terms... For cycles longer than 1, I propose to consider only the smallest integer of the cycle. This way, we will be sure to talk about the same cycle each time. I also intend to make a general table showing the following : For all bases, look at the percentage of the number of sequences that end with all prime numbers up to 1000. But the first column will give percentages for all integers. The only problem is that except for base 2, the number of sequences ending with a prime number is small. Even for base 2, we have only 500 sequences and this is still a statistically low number ! By the way, I confirm you in this post that I found exactly the same thing as you for the prime numbers (or numbers belonging to a cycle) that end the sequences of base 2. Here is the result of running my program, in a convenient form for use with the Sage software : Code:
[3, [2, 4, 55, 164, 305, 317]] [6, [14, 15, 59]] [7, [3, 10, 12, 141, 278, 387, 421]] [11, [60, 316, 480, 499]] [19, [39, 76, 190, 219]] [31, [5, 101, 146]] [37, [68, 125, 243]] [41, [6, 8, 23, 47, 112, 117, 281, 373, 405, 411]] [43, [9, 62, 210, 271]] [53, [20, 78, 214, 347, 450]] [71, [43, 177]] [89, [32, 82]] [97, [51, 73]] [101, [301, 425]] [241, [279, 346]] [547, [24, 198]] [7481, [69, 458]] I'm also working on your idea of (rare) abundant numbers analysis in base 2 sequences. But this work can be generalized to all bases for sequences that end trivially : those n^i whose exponent i and n have the same parity (green cells). Like for base 2, there is no exception, none of these sequences is OpenEnd : in no table for no base, there are three orange cells following each other ! Last fiddled with by garambois on 20200707 at 08:19 Reason: For greater clarity 
20200707, 13:41  #325  
"Ed Hall"
Dec 2009
Adirondack Mtns
2×1,627 Posts 
JeanLuc,
I can provide all the tables in a "relatively" short time, but I was waiting to see what other data might be of interest before completing a full run. I have some additions I'm working with. Your extra items should fit right in. Quote:
Is there interest in listings of abundant and deficient lines, or would that just be too much? What about common primes within a sequence? I know there are many trivial primes (2, 3, 5, 7, etc.), but what about a listing of any repeated primes above a particular value? I believe all cycles' .elf files end with the smallest value, but what I'm having trouble with is detecting that that value is part of a cycle. I'll still work on this, but it's moved to lower on my list until I get some of the other things worked out. I'm currently reporting on all cells of a table, at their current status, but I'm not reporting whether the sequence terminates. Is that significant? I think I can add that, but I'll see how troublesome it is. I'll work on a bit of this other stuff and attach some files, hopefully not too far off. 

20200707, 16:09  #326 
"Rich"
Aug 2002
Benicia, California
2×557 Posts 

20200707, 17:04  #327 
"Ed Hall"
Dec 2009
Adirondack Mtns
2·1,627 Posts 
Latest Data Harvesting
Here's of an example of my latest play. I'm showing 8128^1 through 8128^5, because this range shows all three endings (Cycle, Prime, Open):
Code:
Statistics for Sequences in Base 8128 Table Stats for 8128^1: The first term is perfect. Total abundant terms: 0 Total deficient terms: 0 Most sequential abundant terms: 0 Most sequential deficient terms: 0 Parity changes even to odd: 0 Parity changes odd to even: 0 Number of abundant peaks: 0 The complete (current) last line is: 2 . 8128 = 2^6 * 127 Sequence terminates in a cycle. Stats for 8128^2: The first term is abundant. Total abundant terms: 1 Total deficient terms: 0 Most sequential abundant terms: 1 Most sequential deficient terms: 0 Parity changes even to odd: 1 Parity changes odd to even: 0 Number of abundant peaks: 0 The complete (current) last line is: 1 . 67096703 = 67096703 Sequence terminates in a Prime. Stats for 8128^3: The first term is abundant. Total abundant terms: 31 Total deficient terms: 44 Most sequential abundant terms: 11 Most sequential deficient terms: 30 Parity changes even to odd: 1 Parity changes odd to even: 0 Number of abundant peaks: 7 Greatest abundant peak (13dd): 8188451842630 Greatest peak larger than first term: Yes The complete (current) last line is: 75 . 588169 = 588169 Sequence terminates in a Prime. Stats for 8128^4: The first term is abundant. Total abundant terms: 1 Total deficient terms: 6 Most sequential abundant terms: 1 Most sequential deficient terms: 6 Parity changes even to odd: 1 Parity changes odd to even: 0 Number of abundant peaks: 1 Greatest abundant peak (16dd): 4433780393574655 Greatest peak larger than first term: Yes The complete (current) last line is: 7 . 4676239 = 4676239 Sequence terminates in a Prime. Stats for 8128^5: The first term is abundant. Total abundant terms: 668 Total deficient terms: 235 Most sequential abundant terms: 243 Most sequential deficient terms: 27 Parity changes even to odd: 0 Parity changes odd to even: 0 Number of abundant peaks: 30 Greatest abundant peak (108dd): 132753974028868296806704376876902835792862284557288443803825213072201704392301945780093329848100479057972132862079756272 Greatest peak larger than first term: Yes The complete (current) last line is: 903 . 132753974028868296806704376876902835792862284557288443803825213072201704392301945780093329848100479057972132862079756272 = 2^4 * 3 * 11 * 31 * 73 * 6703 Sequence is still open. Last fiddled with by EdH on 20200707 at 17:07 Reason: misspillin repar 
20200707, 18:24  #328 
"Ed Hall"
Dec 2009
Adirondack Mtns
3254_{10} Posts 
Here's a current run for the base 2 table:
Edit: Added bases 3, 5 and 6: Edit2: Added base 7: Edit3: I have noticed a flaw! For open sequences that are currently riding an abundant climb, I have considered the last term as the current maximum if it is higher than the others. However, the size (xxdd) represents the previous maximum. I will fix this and reattach the affected files. Edit4: Files replaced with corrected versions. (Files current as of Edit date shown below.) Last fiddled with by EdH on 20200708 at 01:59 
20200708, 09:51  #329 
Oct 2011
5·53 Posts 
@ richs :
OK, I'll make all the changes in the next update, very soon... A lot of thanks ! @ EdH : Thank you very much for all your files. Bases 2, 3, 5, 6 and 7 should do me for now. I will try to verify all of this and try to reproduce the results of your program in a few days. Because now, I'm focusing on the sequences that end with prime numbers and creating a table to compare the bases between them. I think I will be able to show this table here in this forum today or tomorrow... I have a suggestion for the presentation of your results. If it is not too complicated for you, would it be possible to recreate the files for bases 2, 3, 5, 6 and 7 by doing the modification like this : Replace this : Code:
Stats for 3^5: The first term is deficient. Total abundant terms: 1 Total deficient terms: 6 Most sequential abundant terms: 1 Most sequential deficient terms: 4 Parity changes even to odd: 2 Parity changes odd to even: 2 Number of abundant peaks: 1 Greatest abundant peak (2dd): 16 Greatest peak larger than first term: No The complete (current) last line is: 7 . 3 = 3 Sequence terminates in a Prime. Code:
[3, 5, 0, 1, 6, 1, 4, 2, 2, 1, 16, 0, 7, 3] [a, b, c, d, e, f, g, h, i, j, k, l, m, n] a : base b : exponent c : 0 if deficient, 1 if abundant d : number of total abundant terms e : number of total deficient terms f : most sequential abundant terms g : most sequential deficient terms h : parity changes even to odd i : parity changes odd to even j : number of abundant peaks k : greatest abundant peak (2dd) l : greatest peak larger than first term 1 if yes and 0 if no m : last index n : last term There's no problem if it's too constraining for you to recreate the files like that, I'd settle for the original form ! And to answer some of your questions above :  In my opinion, yes, a single abundant line is of interest.  Yes, I plan to work in August on all the prime numbers that appear in a sequence and not only on the last one. I'll take into account even the small prime numbers...  There are so few cycles at the moment in our data that I plan to fix the problem by writing the data manually for only the cycles !  The work on openend sequences will only be useful when I work on the prime numbers appearing in all the terms of the sequence and not only on the last terms of the sequences that end. 
20200708, 11:19  #330 
"Ed Hall"
Dec 2009
Adirondack Mtns
2·1,627 Posts 
This should be fairly simple. I have the programs/scripts written and .elfs saved. I just need to modify some things.
I'll skip open sequences for now, since you mention it for later work. That way I won't have to check for updates of any .elfs. For later, as to primes found within a sequence, will you need a count of each or just a list? 
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