Aliquot sequences that start on the integer powers n^i

[EdH]

Originally Posted by garambois

Thank you very much.
There will be even more colors in the update in a few days !

I look forward to seeing all the colors!

Are there more data points for consideration?

A couple things interesting to me:

1. ATM, 2^544 has a "3" downdriver.

2. 2^543 shed 69 digits from index 9 to index 10 (151-82 dd).

[garambois]

Originally Posted by EdH

I look forward to seeing all the colors!

Are there more data points for consideration?

A couple things interesting to me:

1. ATM, 2^544 has a "3" downdriver.

2. 2^543 shed 69 digits from index 9 to index 10 (151-82 dd).

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...

[EdH]

@Jean-Luc: 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?

[garambois]

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 !

[richs]

Originally Posted by richs

Reserving 439^28 at i407.

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.

[garambois]

OK, page updated.
Many thanks to all...

[RichD]

30^43 has surely merged. No way I could get that sequence to a C158 is such a short time.

[EdH]

Originally Posted by RichD

30^43 has surely merged. No way I could get that sequence to a C158 is such a short time.

30^43:i2207 merges with 39060:i2

[garambois]

OK, thanks, I'll add this merger in the next update !

[garambois]

Originally Posted by garambois

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.
Sequences that start on integer powers seem to generally end with the same probability on the same prime numbers as all of the sequences.
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 !

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 !

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...

[unconnected]

I'm done with n=13, all sequences after 13^80 now are >120 digits with >110 digits composites passed ECM work.