Aliquot sequences that start on the integer powers n^i

[Happy5214]

Originally Posted by garambois

If you look at conjectures 134, 135, 136 and 140 on this page dedicated to conjectures related to this project, they all talk about bases that are primorial numbers.
That's why I think we can say that the primorial bases play a special role in our project, so they are "special".
In principle, we can also add a category for bases that are factorials when we will have several of them, why not ?
None of these bases can be taken out of this category in the future, no matter what.

My bad, thanks for that reminder (of a page I helped you create).

Originally Posted by garambois

If you have additional computational resources, you can complete the calculations for some trivial sequences of bases 210, 14316, 14264, 14536 or other still unreserved bases.
Otherwise, you can also advance the non-trivial sequences of these unreserved bases to for example 105 or 110 digits. With a little luck, you will finish one or other non-trivial sequence this way.
But in any case, please report here any work you undertake : it would be too bad if several of you were to calculate the same sequence !
And don't forget to tell me about the new initialized bases, I will include them in the general scan on Monday, even if these sequences are not yet added to our project page.

Um, I know I haven't made much progress on base 210, but I haven't released it yet, nor have I been informed that my reservation has been canceled. You're free to work on the pink sequences, but I want dibs on the trivial orange ones.

[garambois]

Originally Posted by RichD

Bases 52, 54 and 55 should have all exponents (terms) initialized to at least 80 digits and all 80 digit or less trivial terminating sequences completed. Many will be advanced further by late Monday. I'll put emphasis on expected terminating sequences and raise the initial size to 100 (or more) digits in the next couple days.

OK, this is excellent.
I will add all these bases in my list to scan monday.

Originally Posted by Happy5214

Um, I know I haven't made much progress on base 210, but I haven't released it yet, nor have I been informed that my reservation has been canceled. You're free to work on the pink sequences, but I want dibs on the trivial orange ones.

Your reservation for base 210 was not cancelled, I would never have done that without asking your permission !
But when I had extended the base (added more exponents), I did not feel entitled to reserve these new sequences for you without your request.
So the base 210 is "officially" reserved for you until exponent 52.
But I understand your request, and in the next update, I will extend your reservation to exponent 59, i.e. I will add all the orange cells to your reservation.


Note :
We find this same phenomenon of "partially reserved base" following an extension of the number of exponents for many other bases.
For example, the bases : 13 (VBC), 24 (HPP), 28 (LRV), ...
There are others and even some of mine, like the base 6469693230 which is reserved for me only up to exponent 14.
I don't have the computing power to go beyond that !
I am currently calculating the sequence 6469693230^14 and other bases that are disproportionately large for me, such as 223092870^15 (I am at index 15 and an NFS decomposition is underway on my computer for a C144 since July 21 !)

[Aillas]

Originally Posted by garambois

Thanks a lot.
But forgive my ignorance, I can't compile this program under Linux, here is what my compiler says :

garambois@floyd:~/sage-9.2$ g++ Alimerge.cpp -o Alimerge
Alimerge.cpp: In function ‘int main(int, char**)’:
Alimerge.cpp:67:5: error: ‘sscanf_s’ was not declared in this scope
     sscanf_s(argv[1], "%d", &base);
     ^~~~~~~~
Alimerge.cpp:67:5: note: suggested alternative: ‘sscanf’
     sscanf_s(argv[1], "%d", &base);
     ^~~~~~~~
     sscanf
  garambois@floyd:~/sage-9.2$

Please, can you help me ?

I think an include is missing. According to the doc I will say it should work if you add

#include <stdio.h>

I will try to find a Linux to test it...

[Aillas]

New version that compile on my unix
I need to force c++11
I also replace sscanf_s by sscanf

g++ Alimerge.cpp -std=c++11 -o Alimerge

[EdH]

Originally Posted by Aillas

New version that compile on my unix
I need to force c++11
I also replace sscanf_s by sscanf

g++ Alimerge.cpp -std=c++11 -o Alimerge

It seems to compile fine on an Ubuntu machine with just:

$ g++ Alimerge.cpp -o Alimerge

However, it won't retrieve the txt file (even after I installed curl):

The 80 digit file was not found - download it? (y/n): y
sh: 1: curl -q -s -o OE_3000000_C80.txt http://www.aliquotes.com/OE_3000000_C80.txt: not found
Trouble has occurred while trying to read the 80 digit file!

Also, it doesn't seem to run the full range of exponents:

$ ./Alimerge 45 10 30
Running base 45 from 10 through 30 . . .
Downloading base 45^10 : Done
Found the last 80 digit composite in base 45: [72709411053662444980210157745602152977060291142818210791593789587027512665674740]
80 digit composite has a matching in base 427600
Downloading base 427600 : Done

45^10:i1319 merges with 427600:i0
Downloading base 45^11 : Done
Could not find a 80 digit composite in base 45

versus:

$ ./alimerge3 45 1 30
Running base 45 from 1 through 30 . . .
45^10:i1319 merges with 427600:i0
45^14:i213 merges with 260502:i60
45^24:i448 merges with 39852:i5
Run took 104 seconds.

Thanks for your help!

[Aillas]

Originally Posted by EdH

It seems to compile fine on an Ubuntu machine with just:

$ g++ Alimerge.cpp -o Alimerge

However, it won't retrieve the txt file (even after I installed curl):

The 80 digit file was not found - download it? (y/n): y
sh: 1: curl -q -s -o OE_3000000_C80.txt http://www.aliquotes.com/OE_3000000_C80.txt: not found
Trouble has occurred while trying to read the 80 digit file!

Also, it doesn't seem to run the full range of exponents:

$ ./Alimerge 45 10 30
Running base 45 from 10 through 30 . . .
Downloading base 45^10 : Done
Found the last 80 digit composite in base 45: [72709411053662444980210157745602152977060291142818210791593789587027512665674740]
80 digit composite has a matching in base 427600
Downloading base 427600 : Done

45^10:i1319 merges with 427600:i0
Downloading base 45^11 : Done
Could not find a 80 digit composite in base 45

versus:

$ ./alimerge3 45 1 30
Running base 45 from 1 through 30 . . .
45^10:i1319 merges with 427600:i0
45^14:i213 merges with 260502:i60
45^24:i448 merges with 39852:i5
Run took 104 seconds.

Thanks for your help!

Sorry for the inconvenience. I didn't test for a range of exponents.
All the issues are fixed now.
- Fix the URL to Download of the OE_3000000_C80.txt
- If an C80 was not found for an exponent, the application was stopped.
- Remove output. It was too verbose when checking multiple exponents.

I tested with your example:

./Alimerge 45 1 30
Running base 45 from 1 through 30 . . .
45^10:i1319 merges with 427600:i0
45^14:i213 merges with 260502:i60
45^24:i448 merges with 39852:i5

Total running time   : 1 Minutes : 39 Seconds : 174 Milliseconds (99 seconds.)
Downloading file time: 1 Minutes : 38 Seconds : 770 Milliseconds
Computation only time: 403 Milliseconds

[EdH]

Originally Posted by Aillas

Sorry for the inconvenience. I didn't test for a range of exponents.
All the issues are fixed now.
- Fix the URL to Download of the OE_3000000_C80.txt
- If an C80 was not found for an exponent, the application was stopped.
- Remove output. It was too verbose when checking multiple exponents.

I tested with your example:

./Alimerge 45 1 30
Running base 45 from 1 through 30 . . .
45^10:i1319 merges with 427600:i0
45^14:i213 merges with 260502:i60
45^24:i448 merges with 39852:i5

Total running time   : 1 Minutes : 39 Seconds : 174 Milliseconds (99 seconds.)
Downloading file time: 1 Minutes : 38 Seconds : 770 Milliseconds
Computation only time: 403 Milliseconds

Excellent! Thanks for the quick reply.

All seems to work well now, although "computation only time" is affected by downloading the txt file:

$ ./Alimerge 45 10 30
The 80 digit file was not found - download it? (y/n): y
Running base 45 from 10 through 30 . . .
45^10:i1319 merges with 427600:i0
45^14:i213 merges with 260502:i60
45^24:i448 merges with 39852:i5

Total running time   : 1 Minutes : 42 Seconds : 232 Milliseconds (102 seconds.)
Downloading file time: 1 Minutes : 43 Seconds : 948 Milliseconds
Computation only time: -1 Seconds : -715 Milliseconds

Of course, this only happens if the txt file is missing.

Thanks for all your help.

[garambois]

OK, thank you very much !
I manage to compile and everything seems to work fine now !
When I add new bases to the project, I use this program (Edwin's version so far) : I save a lot of time this way, because if I had to search for the merges manually, it would be very very long !

./Alimerge 12 1 150
Running base 12 from 1 through 150 . . .
12^15:i294 merges with 3366:i2
12^19:i120 merges with 23324:i10
12^29:i790 merges with 1778224:i0
12^35:i380 merges with 3876:i5
12^43:i488 merges with 2484:i8
12^47:i1301 merges with 1374120:i189

Total running time   : 7 Minutes : 20 Seconds : 510 Milliseconds (440 seconds.)
Downloading file time: 7 Minutes : 20 Seconds : 33 Milliseconds
Computation only time: 477 Milliseconds

[garambois]

I continued to complete the page that summarizes all the conjectures stated through this project.

1) I added all the demonstrations for the "trivial" conjectures that were missing (presence of a prime number at index 1 for some exponents for bases that are prime numbers).
2) I corrected some errors.
3) I corrected the statement of the conjecture (79) : base 7 instead of 5.
4) I invalidated the conjecture (81) in contradiction with the conjecture (82), which is maintained. It must have been an error of formulation.
5) The conjectures (130)* and (132)* were classified as two stars instead of one, this was an error. Note that from conjecture (134) onwards, we no longer used this classification with stars, which was supposed to give an idea of the difficulty to prove them.
6) The conjectures (82), (103) and (116) were really very "obvious". Can you just tell me if this formulation (in red) is correct in English language, please : s(7k) = 1 + 7¹ + 7² + ... + 7^k-1 = 1 modulo 7.

Please let me know if you find an error in the proofs.

[Happy5214]

Initialization on base 564 is done. I also did the trivial sequences under 70 digits (and a couple past) for all of the remaining three-digit bases on https://oeis.org/A131884. One termination at 6 and a few at 3 were found, but it was mostly large primes.

210^52 and 210^54 also have been terminated with primes.

Edit: Conjecture (82) has a rather trivial proof: s(7^k) = (7^k-1)/6, and since 7^k-1 can never be divisible by 7 (being one less than a power of 7), neither can (7^k-1)/6. Similar proofs apply for the other prime bases.

[garambois]

Originally Posted by Happy5214

Edit: Conjecture (82) has a rather trivial proof: s(7^k) = (7^k-1)/6, and since 7^k-1 can never be divisible by 7 (being one less than a power of 7), neither can (7^k-1)/6. Similar proofs apply for the other prime bases.

Yes, we wrote our posts at the same time, I added point 6) in my post #1307 !
Can you just tell me if the word "modulo" is correct please and if the proofs of conjectures 82, 103 and 116 are written in an acceptable way ?