Aliquot sequences that start on the integer powers n^i

[garambois]

Originally Posted by EdH

@Jean-Luc:
Now that we have a clearer picture of all the added tables, is there something in particular that you would prefer I work on?

If not, I'll go ahead and dabble in the base 79 table for a bit - no reservations necessary, for now.

Thank you for all the work keeping the tables straight!

This time, I had about 3 - 4 hours of work to do the update !

It is likely that at the next update, I won't touch any more the bases where nobody reserved sequences. I will update these bases only once every 2 or 3 months.

I think that the best thing for the moment is indeed to continue the calculations of the different bases that we have started. We will see how much new data we will have in a few weeks, months, or even a year.
It all depends on how many volunteers are going to do the calculations.

But everyone should be well aware that we may not find what we were looking for at the beginning. This project was born in the following way. When with friends we were looking at the infinite graph of the aliquot sequences (see the graph here), we wondered if among the sequences starting with integer powers of a prime number, there would be more that would end with the same prime number. For example, would sequences starting with 7^i (i=1, 2, 3...) end more often with the prime number 7 ?
Today, we cannot answer this question.
But we have found something else that we were not looking for at all at the beginning : the 133 conjectures of post #447. One never knows to which ideas the chance of reflections can lead us ?
Maybe all the calculations we make will allow us to answer the original question. Maybe not ! Or maybe we'll again find something else that we didn't expect to find.
But to find things, we have to look at the data and ask ourselves all sorts of questions. Anyone can do that... You have to be as creative as possible.

But right now, I'm still waiting for an Open-End sequence in base 2, 18 or another base with an exponent that has the same parity as the base.

[EdH]

@Jean-Luc:
Thanks for all the work and explanation. I will possibly play with extending base 18, if RichD isn't working there anymore and then play randomly with the matched parity unreserved sequences in the various tables.

But, as we study these, I'm beginning to get a different feeling for "open" sequences. Open really only means we haven't finished it. 2^552, after index 0, had 9 abundant indices. If, when it reached 168 digits, we stopped, it would have been considered "open" and would have met the search criteria.

@RichD:
Are you still doing work on base 18?

[RichD]

Originally Posted by EdH

@RichD:
Are you still doing work on base 18?

All done with n=18. It's all yours.

I am slowing down on the others but still randomly doing a bit of work on n=19, 20, 23, 29, 770.

[VBCurtis]

Originally Posted by EdH

But, as we study these, I'm beginning to get a different feeling for "open" sequences. Open really only means we haven't finished it.

This is the original hypothesis of the Aliquot-sequence project- that they all terminate, eventually. We keep cracking C190+ cofactors in hopes of breaking a driver and finding a new largest-peak-size-that-still-terminates record.

[EdH]

Originally Posted by RichD

All done with n=18. It's all yours.
. . .

Thanks - I'll work there for a while, then.

Originally Posted by VBCurtis

This is the original hypothesis of the Aliquot-sequence project- that they all terminate, eventually. We keep cracking C190+ cofactors in hopes of breaking a driver and finding a new largest-peak-size-that-still-terminates record.

When I first started playing with Aliquot sequences, I was in the Guy-Selfridge camp. But, since I've been working on this sub-project, I'm swinging much more toward the Catalin-Dickson Conjecture. However, I have to keep in mind that we are really working on a tiny piece of the "big" picture by working with bases and powers. Then again, if all bases and powers terminate, then everything does, because all numbers can be bases.

[garambois]

Originally Posted by EdH

But, as we study these, I'm beginning to get a different feeling for "open" sequences. Open really only means we haven't finished it. 2^552, after index 0, had 9 abundant indices. If, when it reached 168 digits, we stopped, it would have been considered "open" and would have met the search criteria.

For 2^552, yes : 9 abundant indices.
But not 9 even terms !

Just for information : I'm in the Guy-Selfridge camp and I think that one day we'll be able to build a number that we'll be sure is the start of an Open-End sequence that will grow endlessly...

[EdH]

Originally Posted by garambois

For 2^552, yes : 9 abundant indices.
But not 9 even terms !

Just for information : I'm in the Guy-Selfridge camp and I think that one day we'll be able to build a number that we'll be sure is the start of an Open-End sequence that will grow endlessly...

Ah, so we need even terms for it to be open, as well, because odd terms will eventually descend?

I won't hold that against you. I might even move back that way myself. But, currently, we seem to be displaying a lot of evidence for terminations.

[EdH]

Originally Posted by VBCurtis

This is the original hypothesis of the Aliquot-sequence project- that they all terminate, eventually. We keep cracking C190+ cofactors in hopes of breaking a driver and finding a new largest-peak-size-that-still-terminates record.

But, we need some new algorithms. My attention span is short. I can't reasonably factor a difficult c190.

[VBCurtis]

Originally Posted by EdH

I can't reasonably factor a difficult c190.

That's what nfs@home is for. We merely need to ECM-pretest and solve the matrix, a total of at most 20% of the sieve time.

Edit: Of course, all that does is delay the onset of not-enough-attention-span to ~C200.

[garambois]

Originally Posted by EdH

Ah, so we need even terms for it to be open, as well, because odd terms will eventually descend?

I won't hold that against you. I might even move back that way myself. But, currently, we seem to be displaying a lot of evidence for terminations.

In fact, I don't really know what we should consider an Open-End sequence. But I do make two observations :

1) In the main project, on the blue page, I've never seen an Open-End sequence that starts on an odd number. But maybe there is one and I don't know how to filter it ?

2) On all the aliquots sequences that start on integers from 1 to 10,000,000, thanks to my fundamental database, I know that the absolute record for a growth sequence with consecutive odd terms is held by the sequence that starts on the integer 2,551,185. And this number of growing consecutive terms is only 6, which is very little. (see here some other records)

Having made these two observations, I tell myself that a sequence can only be Open-End with even terms for numbers of a few hundred digits that we work with.
But maybe it should also be possible to have an Open-End sequence with huge odd terms. But discovering such odd Open-End sequences (if they exist) is far from being within our reach.

[EdH]

In case you do an update this weekend prior to my finishing 18^119, I have otherwise completed a new row for base 18, through 18^118 at present.

I am running 18^119 and hope to have it done soon. For now, I will not be extending the table further.