Aliquot sequences that start on the integer powers n^i

[EdH]

Originally Posted by garambois

Excellent !
This is very interesting !
I understand that this is only a very small sample !
I will write the same program in early summer and compare our results.
I can't wait to see the full file and examine this data.
I'm already asking myself a few questions just by seeing this little sample :

1) For each prime number, do we always have one of the two bases that is necessarily base 2 ? (I think this is just due to the way the program is written and the fact that the sample is small ?)

2) Do you also test primes larger than 10 digits ?

3) Do you test all the bases listed on the project page ? Have we then overestimated the computation time that this test would take ?

4) Is this program written in C, python or another language ?

What I listed was a few lines from an output of one of my tests. Oddly, a complete base 2 run only yielded one more prime that spanned multiple bases, so I'm not sure how I made a grab of the perfect part of an overall list to show what looked like better results. Part of the trouble was that my overall listing also included numbers where the prime digits were within a larger prime. I have since been able to omit such cases. But, I still don't see how my sample was so close to the whole set.

1) The list was limited to the primes found within the base 2 sequences. This will be expanded later, but I will need to find a way to avoid total duplication of finds.

2) This test is running everything greater than 9 digits.

3) Currently, I'm using all bases through 200, except for the recently added. I haven't run a full update in a while, due to the db limitations.

4) It's a mixture, but mostly bash scripts, with one C++ program that harvests the primes.

ATM, I am becoming totally confused! I'm going to take a break until I sort out what I'm doing.

[garambois]

OK, thanks Edwin for those answers.
In any case, all the sample data seems to match except for one.
I can't find the prime number 2159188693 at index 18 of sequence 2^327.

Let us know if you do more tests in the next time and if you find other "big" prime numbers that are present in several bases.
I will look into this.
We never know, we have to check, even if I think that all this is only a coincidence and that we will not have much larger primes being in two different bases.
It would surprise me a lot if there were such prime numbers of 12 digits or more !

As far as I'm concerned, I have one fear : I write my programs in python and I'm afraid it's infinitely slower than what you write in C.
We'll see in July, because I unfortunately don't have enough time at the moment !

[EdH]

Originally Posted by garambois

. . .
I can't find the prime number 2159188693 at index 18 of sequence 2^327.
. . .
As far as I'm concerned, I have one fear : I write my programs in python and I'm afraid it's infinitely slower than what you write in C.
We'll see in July, because I unfortunately don't have enough time at the moment !

The problem with cleaning things up manually for posting - that should be 2^372:18. I have made that part of the scripts now, and below is another set, with your requested formatting. I didn't verify anything, but something seems very wrong with this listing! I'm going to let it sit for now and try for a clearer "big picture," maybe tomorrow.

As for the programming, the C++ program is a very small part and most of the time is spent in the scripts. I hope to do a lot of work refining how the data are collected. I'll also need to do another full update of all my local sequences. I did add all the new bases below 200, so when I resume work, they will be represented.


Latest run:

1051654267,19,22,2102
1051654267,2,468,58
1097038783,2,501,61
1097038783,67,83,11
1129552253,19,18,510
1129552253,2,341,25
1639132051,22,41,508
1639132051,2,422,11
2159188693,2,372,18
2159188693,50,98,7
3206363219,2,448,19
3206363219,2,516,53
3206363219,59,36,1559
3258313481,2,305,11
3258313481,24,17,1520

[garambois]

Originally Posted by EdH

The problem with cleaning things up manually for posting - that should be 2^372:18. I have made that part of the scripts now, and below is another set, with your requested formatting. I didn't verify anything, but something seems very wrong with this listing! I'm going to let it sit for now and try for a clearer "big picture," maybe tomorrow.

As for the programming, the C++ program is a very small part and most of the time is spent in the scripts. I hope to do a lot of work refining how the data are collected. I'll also need to do another full update of all my local sequences. I did add all the new bases below 200, so when I resume work, they will be represented.


Latest run:

1051654267,19,22,2102
1051654267,2,468,58
1097038783,2,501,61
1097038783,67,83,11
1129552253,19,18,510
1129552253,2,341,25
1639132051,22,41,508
1639132051,2,422,11
2159188693,2,372,18
2159188693,50,98,7
3206363219,2,448,19
3206363219,2,516,53
3206363219,59,36,1559
3258313481,2,305,11
3258313481,24,17,1520

Thank you very much Edwin for all this work.
The format is perfect like this.

I can still spot an error : the prime number 3206363219 doesn't seem to appear anywhere in the sequence 2^448.
I immediately checked this prime, because it appears in three lines of the list and 516 is not a multiple of 448 and anyway, the indexes are > 1.
So it was very intriguing and very unlikely that this prime would appear in 3 lines.

Edwin, you are already doing a lot for this whole project !
Only take on this extra work if you find interest in it !
And in any case, take your time, I won't get serious about this until after July 7 - 10 anyway.
But I must admit that you have discovered a very interesting path to explore in my humble opinion.
It's risky to go into this alone, there are many sources of error.
It is very stimulating to work on such an idea with several people and especially, so, we have means to verify.
Thanks Edwin !

[EdH]

Originally Posted by garambois

. . .
I can still spot an error : the prime number 3206363219 doesn't seem to appear anywhere in the sequence 2^448.
I immediately checked this prime, because it appears in three lines of the list and 516 is not a multiple of 448 and anyway, the indexes are > 1.
So it was very intriguing and very unlikely that this prime would appear in 3 lines.
. . .

I must have grabbed this from an earlier run. This was part of my original problem:

base2/2^448.elf:19 .   15006007375820463402452561600162328700320636321975920107434993415977015910414828098588984394591 = 17 * 2389 * 2016739 * 141986827 * 190519331 * 6772722565336945433810924560372709070485550949679612135489377339249

I think I have that fixed. I plan to run base 2 again in a little while. I'll see if 2^448 is absent in those results

[garambois]

OK, I understand the problem much better with this example, thanks.

[chris2be8]

My first cut would be:

Put all primes into a flat file, called primes.txt below.
sort <primes.txt | uniq --repeated >primes.duplicates

That should get a list of all primes that appear more than once in the list. Then see how many there are and decide how to process them.

[EdH]

Thanks Chris! I will add this into my study.

[garambois]

Yes, many thanks Chris.
I think you are right and the program will be more effective this way.
Perhaps even incomparably more effective !

[EdH]

@Chris: uniq is definitely helping!

Another wrinkle has made itself known, that I will have to look into:

If you check prime 989948318349327032515056706609970813346117930160196095662320313189860502565627388119428970284861410132787555691, you will find six occurrences across as many sequences:

3^10:3460
5^26:3674
24^57:5346
47^97:3551
70^3:3478
85^2:3467

But, it's because they all merge with 1134. Additionally, all other primes in those terms from then on will be multiple hits as well:

2^4
7
31
107
1747
32869
151597
28936680745039

[garambois]

Wow, wow, wow !
It's going to be a lot more complicated than I thought if we have to remove the multiple cases due to mergers.
I hadn't thought of that.
For the moment, as far as I am concerned, I do not see how to deal with this problem !