Aliquot sequences that start on the integer powers n^i - Page 35

EdH · 2020-07-30, 21:02

Quote:

Originally Posted by EdH

. . .
I am currently doing all the preliminary work for a table to be added for 2310. I'm not sure if I will color in the transparent cells or not (like before with 30030).

All the preliminary work is done for table 2310 (opens at 100* or better dd, all matched parity terminated with primes). There is one merge:

Code:

2310^1:i1 merges with 1578:i4

*ECM was not performed on final composites.

richs · 2020-07-30, 21:43

Quote:

Originally Posted by richs

Reserving 439^26 at i373.

439^26 is now at i515 (added over 140 iterations) and a C131 level with a 2^2 * 3 * 7 guide, so I will drop this reservation. The remaining C120 term is well ecm'ed and is ready for NFS.

Reserving 439^28 at i407.

Happy5214 · 2020-07-31, 12:00

n=24 is done to i=20, and I'm releasing the sequences below that limit. Next, I'll bring n=21, i=70 to 80 up to C120 co-factors.

garambois · 2020-08-03, 09:32

Sorry for the late update, but I just came back from my vacation...
Page updated.
Thank you all for your hard work !
I ask you to please check if the updates concerning you are correct ?

@EDH : I think we need to modify merges for 2310 and 30030.

Code:

2310^1:i1 merges with 1578:i4
30030^1:i1 merges with 22518:i4

Should be :

Code:

2310^1:i0 merges with 1578:i3
30030^1:i0 merges with 22518:i3

Do you confirm that this change is correct ?

I will now be able to continue quietly to examine all the accumulated data. I will be refining and running my analysis algorithms over the next few days. I'll keep you informed if a conjecture should arise, hoping this time it's not already known !

EdH · 2020-08-03, 14:14

Welcome back, Jean-Luc! I hope you had a great time!

You are correct about both merges. Now I have to find out why my program wasn't. I seem to recall correcting this already. Maybe I used an earlier (incorrect) version somehow.

I stumbled around with a couple things in post 364 that might be of interest. I'm sure I don't have it written out properly, but I don't think the very last one is something that will actually turn up via data review, but it might just be a form of what you already found:

Code:

Additionally, that aⁱ+1 is a factor of s(a^(i*n)) (n, a positive even integer)

Basically, if I have this correct, aⁱ+1 will divide evenly, the Aliquot sum of a^(i*n), if n is a positive even integer.

Of course, I might be way off with something.

garambois · 2020-08-03, 16:07

Yes, I did see your posts #364 and #371 and I'll look into it.
But it's going to take me a few days to run the data analysis programs.
The execution times are very long and I don't understand why.
I'll try to shorten these execution times.

I also think that to better notice things, I'll have to modify the program that gives the output :

Code:

base 2    prime 197748738449921    exponent 265
base 2    prime 197748738449921    exponent 530

base 2    prime 242099935645987    exponent 198
base 2    prime 242099935645987    exponent 396

base 2    prime 332584516519201    exponent 19
base 2    prime 332584516519201    exponent 382

In order for the output to become something like this :

Code:

base 2    prime 197748738449921    exponent 265    at index 1
base 2    prime 197748738449921    exponent 530    at index 1

base 2    prime 242099935645987    exponent 198    at index 1
base 2    prime 242099935645987    exponent 396    at index 1

base 2    prime 332584516519201    exponent 19    at index 1
base 2    prime 332584516519201    exponent 382    at index 1

Or something more like your idea that you present in post #371, like this :

Code:

prime 197748738449921 shows up 2 times (265:i1, 530:i1).
prime 242099935645987 shows up 2 times (198:i1, 396:i1).
prime 332584516519201 shows up 2 times (191:i1, 382:i1).

Indeed, for the other bases, one does not always find the repetitions of prime numbers at index 1, and besides, it is generally not at index 1 anymore.
More curious : by manually and laboriously examining (hence the need to automatically examine) the data in your file attached to post #374, it even happens that a large prime number repeats itself twice in two terms at two indexes of the same sequence.
And this may be a coincidence, but I prefer to be sure !
Then I also want to know if large prime numbers repeat more than twice in a single sequence and if so, at which indexes.

EdH · 2020-08-03, 17:18

I will look into adding indices to my prime listings, but for the the smaller primes, it would cause trouble with lengths of the lines. Is there possibly a lower limit I could use. I think your example showed 10^7, possibly?

I can run the prime listing for a single base in a relatively quick fashion with my setup of bash scripts and compiled C++ program. The script divides the search regions into 8 processes to make use of 8 threads on an i7 and then combines the results for the final file. This helps with array sizes that quickly overrun available space in my program.

I'm not sure what other analyses you might be doing. For a little while I was working with finding duplicate primes (>10^6) across the entire set of tables:

Code:

allpbase11:prime 1000117 shows up 1 time(s) (44).
allpbase13:prime 1000117 shows up 1 time(s) (40).
allpbase2:prime 1000117 shows up 1 time(s) (396).

allpbase14:prime 1000847 shows up 1 time(s) (31).
allpbase2:prime 1000847 shows up 1 time(s) (328).
allpbase3:prime 1000847 shows up 1 time(s) (104).

allpbase15:prime 1001123 shows up 1 time(s) (12).
allpbase2:prime 1001123 shows up 1 time(s) (300).

allpbase10:prime 1002511 shows up 1 time(s) (15).
allpbase1155:prime 1002511 shows up 1 time(s) (4).
allpbase2:prime 1002511 shows up 1 time(s) (312).
 allpbase7:prime 1002511 shows up 1 time(s) (70).
. . .

Here's a section for >10^5:

Code:

allpbase10:prime 100049 shows up 1 time(s) (33).
allpbase11:prime 100049 shows up 1 time(s) (24).
allpbase13:prime 100049 shows up 1 time(s) (42).
allpbase14:prime 100049 shows up 2 time(s) (46, 89).
allpbase15:prime 100049 shows up 1 time(s) (12).
allpbase2:prime 100049 shows up 1 time(s) (303).
allpbase21:prime 100049 shows up 1 time(s) (28).
allpbase385:prime 100049 shows up 1 time(s) (4).
allpbase6:prime 100049 shows up 2 time(s) (101, 135).
allpbase7:prime 100049 shows up 1 time(s) (32).

allpbase10:prime 100057 shows up 2 time(s) (33, 95).
allpbase11:prime 100057 shows up 1 time(s) (42).
allpbase12:prime 100057 shows up 1 time(s) (67).
allpbase15:prime 100057 shows up 1 time(s) (12).
allpbase17:prime 100057 shows up 1 time(s) (12).
allpbase2:prime 100057 shows up 1 time(s) (375).
allpbase210:prime 100057 shows up 1 time(s) (25).
allpbase3:prime 100057 shows up 2 time(s) (204).
allpbase5:prime 100057 shows up 1 time(s) (16).
allpbase510510:prime 100057 shows up 1 time(s) (3).
allpbase6:prime 100057 shows up 1 time(s) (81).
allpbase7:prime 100057 shows up 3 time(s) (18, 96).

allpbase11:prime 100103 shows up 1 time(s) (48).
allpbase14:prime 100103 shows up 2 time(s) (43, 81).
allpbase2:prime 100103 shows up 1 time(s) (346).
allpbase496:prime 100103 shows up 1 time(s) (19).
allpbase6:prime 100103 shows up 4 time(s) (29, 73, 81).
allpbase7:prime 100103 shows up 2 time(s) (36, 50).
. . .

(All the blank lines were added after for readability.)

EdH · 2020-08-03, 20:46

I have added unique* index references. Here is a sample from base2primes:

Code:

prime 162259276829213363391578010288127 shows up 5 times (107:i1, 214:i1, 321:i1, 428:i1, 535:i1).
prime 163537220852725398851434325720959 shows up 4 times (133:i1, 266:i1, 399:i1, 532:i1).
prime 1282816117617265060453496956212169 shows up 2 times (247:i1, 494:i1).
prime 2679895157783862814690027494144991 shows up 3 times (145:i1, 290:i1, 435:i1).
prime 4982397651178256151338302204762057 shows up 2 times (231:i1, 462:i1).
prime 73202300395158005845473537146974751 shows up 2 times (235:i1, 470:i1).
prime 383725126655170964501315730676446647 shows up 2 times (263:i1, 526:i1).

If you would like, I can provide a full set composed of all the current tables from the table pages.

*Unique means that I do not list the same index of an exponent more than once, so if there is a prime with a power, it is listed only once. e.g. 2^7 at index 12 of a particular sequence would be listed as 2:i12 even though there were seven 2s represented. If the prime occurs on a subsequent index it is listed. The count (X times) still represents the total.

garambois · 2020-08-04, 07:52

Yes, this is exactly the program I have in mind : a program that allows you to see unique indexes.
Your program execution time is much faster than mine.
So I'm very interested in your new tables with indexes, like in your last post.
But I finally have an idea that will allow me to greatly reduce the data analysis time and I will soon be able to quickly reproduce your calculations.
See my next post to understand what I'm looking for...

garambois · 2020-08-04, 08:42

I think I've come up with a new conjecture again.
I don't know if it's already known, please let me know if it is ?

This new conjecture concerns on the other hand only one prime number for base 3, but what is new is that we have the presence of this prime number always in the decomposition of two consecutive terms of the sequence !
Here is the statement of the conjecture :
For any aliquot sequence starting with a number of the form 3^(26*k), k integer, the prime number 398581 always appears in the decomposition of the terms of index 1 and index 2.

This is a small conjecture which concerns only a particular case, but perhaps more general conjectures could be found.

To find this, I proceeded as follows :

1) I found this line in the EdH tables :

Code:

prime 398581 shows up 18 times (26, 52, 78, 104, 130, 156, 182, 208, 234).

18 times and only 9 exponents, this is not "usual" ! 18/9=2.

2) I ran a program that makes the indexes appear and I saw this :

Code:

base 3    prime 398581    exponent 26 at index 1
base 3    prime 398581    exponent 26 at index 2
base 3    prime 398581    exponent 52 at index 1
base 3    prime 398581    exponent 52 at index 2
base 3    prime 398581    exponent 78 at index 1
base 3    prime 398581    exponent 78 at index 2
base 3    prime 398581    exponent 104 at index 1
base 3    prime 398581    exponent 104 at index 2
base 3    prime 398581    exponent 130 at index 1
base 3    prime 398581    exponent 130 at index 2
base 3    prime 398581    exponent 156 at index 1
base 3    prime 398581    exponent 156 at index 2
base 3    prime 398581    exponent 182 at index 1
base 3    prime 398581    exponent 182 at index 2
base 3    prime 398581    exponent 208 at index 1
base 3    prime 398581    exponent 208 at index 2
base 3    prime 398581    exponent 234 at index 1
base 3    prime 398581    exponent 234 at index 2

The conjecture then appeared immediately !

I'd like to try and find some more conjectures like that. We have to look at all the bases. But I'm not sure how to write the programs to spot this kind of case !
Maybe for a base, we have to find cases where the number of occurrences of the prime number is a multiple of the number of exponents for which the prime number appears ? In the example above, we have 18/9=2, so we have two indexes per sequence where the prime number 398581 appears and moreover, these indexes are consecutive !
Are there ratios of 3 (3 consecutive or not consecutive indexes), or more ?
Answer in a few days or weeks... And certainly other unexpected things will appear !

Happy5214 · 2020-08-04, 11:35

I've verified this conjecture up to 3^468. 797161 (2*398581-1) also appears in index 1 of all of the sequences I tested with that starting value form. Did you find that in the ones you tested?

2020-08-03, 09:32	#378
garambois "Garambois Jean-Luc" Oct 2011 France 11·59 Posts	Sorry for the late update, but I just came back from my vacation... Page updated. Thank you all for your hard work ! I ask you to please check if the updates concerning you are correct ? @EDH : I think we need to modify merges for 2310 and 30030. Code: 2310^1:i1 merges with 1578:i4 30030^1:i1 merges with 22518:i4 Should be : Code: 2310^1:i0 merges with 1578:i3 30030^1:i0 merges with 22518:i3 Do you confirm that this change is correct ? I will now be able to continue quietly to examine all the accumulated data. I will be refining and running my analysis algorithms over the next few days. I'll keep you informed if a conjecture should arise, hoping this time it's not already known !

2020-08-03, 14:14	#379
EdH "Ed Hall" Dec 2009 Adirondack Mtns 2×19×101 Posts	Welcome back, Jean-Luc! I hope you had a great time! You are correct about both merges. Now I have to find out why my program wasn't. I seem to recall correcting this already. Maybe I used an earlier (incorrect) version somehow. I stumbled around with a couple things in post 364 that might be of interest. I'm sure I don't have it written out properly, but I don't think the very last one is something that will actually turn up via data review, but it might just be a form of what you already found: Code: Additionally, that aⁱ+1 is a factor of s(a^(in)) (n, a positive even integer) Basically, if I have this correct, aⁱ+1* will divide evenly, the Aliquot sum of a^(in), if n is a positive even integer. Of course, I might be way off with something. Last fiddled with by EdH on 2020-09-17 at 14:17*

2020-08-03, 16:07	#380
garambois "Garambois Jean-Luc" Oct 2011 France 11×59 Posts	Yes, I did see your posts #364 and #371 and I'll look into it. But it's going to take me a few days to run the data analysis programs. The execution times are very long and I don't understand why. I'll try to shorten these execution times. I also think that to better notice things, I'll have to modify the program that gives the output : Code: base 2 prime 197748738449921 exponent 265 base 2 prime 197748738449921 exponent 530 base 2 prime 242099935645987 exponent 198 base 2 prime 242099935645987 exponent 396 base 2 prime 332584516519201 exponent 19 base 2 prime 332584516519201 exponent 382 In order for the output to become something like this : Code: base 2 prime 197748738449921 exponent 265 at index 1 base 2 prime 197748738449921 exponent 530 at index 1 base 2 prime 242099935645987 exponent 198 at index 1 base 2 prime 242099935645987 exponent 396 at index 1 base 2 prime 332584516519201 exponent 19 at index 1 base 2 prime 332584516519201 exponent 382 at index 1 Or something more like your idea that you present in post #371, like this : Code: prime 197748738449921 shows up 2 times (265:i1, 530:i1). prime 242099935645987 shows up 2 times (198:i1, 396:i1). prime 332584516519201 shows up 2 times (191:i1, 382:i1). Indeed, for the other bases, one does not always find the repetitions of prime numbers at index 1, and besides, it is generally not at index 1 anymore. More curious : by manually and laboriously examining (hence the need to automatically examine) the data in your file attached to post #374, it even happens that a large prime number repeats itself twice in two terms at two indexes of the same sequence. And this may be a coincidence, but I prefer to be sure ! Then I also want to know if large prime numbers repeat more than twice in a single sequence and if so, at which indexes.

2020-08-03, 20:46	#382
EdH "Ed Hall" Dec 2009 Adirondack Mtns 3838₁₀ Posts	I have added unique* index references. Here is a sample from base2primes: Code: prime 162259276829213363391578010288127 shows up 5 times (107:i1, 214:i1, 321:i1, 428:i1, 535:i1). prime 163537220852725398851434325720959 shows up 4 times (133:i1, 266:i1, 399:i1, 532:i1). prime 1282816117617265060453496956212169 shows up 2 times (247:i1, 494:i1). prime 2679895157783862814690027494144991 shows up 3 times (145:i1, 290:i1, 435:i1). prime 4982397651178256151338302204762057 shows up 2 times (231:i1, 462:i1). prime 73202300395158005845473537146974751 shows up 2 times (235:i1, 470:i1). prime 383725126655170964501315730676446647 shows up 2 times (263:i1, 526:i1). If you would like, I can provide a full set composed of all the current tables from the table pages. Unique means that I do not list the same index of an exponent more than once, so if there is a prime with a power, it is listed only once. e.g. 2^7 at index 12 of a particular sequence would be listed as 2:i12 even though there were seven 2s represented. If the prime occurs on a subsequent index it is listed. The count (X times) still represents the total. Last fiddled with by EdH on 2020-08-03 at 20:49 Reason: clarification*

2020-08-04, 08:42	#384
garambois "Garambois Jean-Luc" Oct 2011 France 11×59 Posts	I think I've come up with a new conjecture again. I don't know if it's already known, please let me know if it is ? This new conjecture concerns on the other hand only one prime number for base 3, but what is new is that we have the presence of this prime number always in the decomposition of two consecutive terms of the sequence ! Here is the statement of the conjecture : *For any aliquot sequence starting with a number of the form 3^(26k), k integer, the prime number 398581 always appears in the decomposition of the terms of index 1 and index 2.** This is a small conjecture which concerns only a particular case, but perhaps more general conjectures could be found. To find this, I proceeded as follows : 1) I found this line in the EdH tables : Code: prime 398581 shows up 18 times (26, 52, 78, 104, 130, 156, 182, 208, 234). 18 times and only 9 exponents, this is not "usual" ! 18/9=2. 2) I ran a program that makes the indexes appear and I saw this : Code: base 3 prime 398581 exponent 26 at index 1 base 3 prime 398581 exponent 26 at index 2 base 3 prime 398581 exponent 52 at index 1 base 3 prime 398581 exponent 52 at index 2 base 3 prime 398581 exponent 78 at index 1 base 3 prime 398581 exponent 78 at index 2 base 3 prime 398581 exponent 104 at index 1 base 3 prime 398581 exponent 104 at index 2 base 3 prime 398581 exponent 130 at index 1 base 3 prime 398581 exponent 130 at index 2 base 3 prime 398581 exponent 156 at index 1 base 3 prime 398581 exponent 156 at index 2 base 3 prime 398581 exponent 182 at index 1 base 3 prime 398581 exponent 182 at index 2 base 3 prime 398581 exponent 208 at index 1 base 3 prime 398581 exponent 208 at index 2 base 3 prime 398581 exponent 234 at index 1 base 3 prime 398581 exponent 234 at index 2 The conjecture then appeared immediately ! I'd like to try and find some more conjectures like that. We have to look at all the bases. But I'm not sure how to write the programs to spot this kind of case ! Maybe for a base, we have to find cases where the number of occurrences of the prime number is a multiple of the number of exponents for which the prime number appears ? In the example above, we have 18/9=2, so we have two indexes per sequence where the prime number 398581 appears and moreover, these indexes are consecutive ! Are there ratios of 3 (3 consecutive or not consecutive indexes), or more ? Answer in a few days or weeks... And certainly other unexpected things will appear !

2020-07-31, 12:00	#377
Happy5214 "Alexander" Nov 2008 The Alamo City 3×5×47 Posts	n=24 is done to i=20, and I'm releasing the sequences below that limit. Next, I'll bring n=21, i=70 to 80 up to C120 co-factors.

2020-08-04, 07:52	#383
garambois "Garambois Jean-Luc" Oct 2011 France 11×59 Posts	Yes, this is exactly the program I have in mind : a program that allows you to see unique indexes. Your program execution time is much faster than mine. So I'm very interested in your new tables with indexes, like in your last post. But I finally have an idea that will allow me to greatly reduce the data analysis time and I will soon be able to quickly reproduce your calculations. See my next post to understand what I'm looking for...

2020-08-04, 11:35	#385
Happy5214 "Alexander" Nov 2008 The Alamo City 3·5·47 Posts	I've verified this conjecture up to 3^468. 797161 (2398581-1) also appears in index 1 of all of the sequences I tested with that starting value form. Did you find that in the ones you tested? Last fiddled with by Happy5214 on 2020-08-04 at 11:38*

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