Aliquot sequences that start on the integer powers n^i

[sweety439]

Originally Posted by garambois

This happens for all bases.

Reminder :
- Sequences whose bases and exponents have the same parity usually terminate trivially (the only exception currently known : 29^15).

I looked the Aliquot sequence of 29^15, this sequence is very interesting, although the first numbers are odd and these numbers decrease quickly, but the sequence reach an odd square number (265^2) and immediately merges with 18528.

I am curious of which number is the smallest odd number whose Aliquot sequence has not yet been fully determined? 1521 = 39^2 is the smallest odd number with long Aliquot sequence.

Edit: I have found it with my program, it is 3025 = 55^2

[bur]

All odd bases with even exponent should switch to an even number with the second term. It would be interesting to see which odd number that isn't a square is of yet open ended.

[Happy5214]

276 and 552 are initialized. 564 is next, where a non-trivial termination is already known (564^5). I'll see if there are any other notable sequences.

[sweety439]

Completed 120^7 at 265 bits (120^2 terminates at 36781, 120^4 terminates at 2643497, 120^6 terminates at 103078882663, 120^8 terminates at 381673, 120^3 and 120^5 merges with other sequences), I will stop only if the number is > 2^256

Now reserving 120^9

[garambois]

Originally Posted by RichD

Looks like somebody might need more numbers to factor. I'll initialize bases 52, 54 & 55 next.

OK, many thanks.
Let me know as soon as I can add these bases to the page.

Originally Posted by richs

392^57 terminates P42 at i27.
392^59 terminates P34 at i84.

Many thanks.
This data will be taken into account in the next update.

Originally Posted by Happy5214

276 and 552 are initialized. 564 is next, where a non-trivial termination is already known (564^5). I'll see if there are any other notable sequences.

Many thanks.
I will add these bases in the next update.

Originally Posted by sweety439

Completed 120^7 at 265 bits (120^2 terminates at 36781, 120^4 terminates at 2643497, 120^6 terminates at 103078882663, 120^8 terminates at 381673, 120^3 and 120^5 merges with other sequences), I will stop only if the number is > 2^256
Now reserving 120^9

Many thanks.
It is not necessary to report all the steps in the initialization of the base 120.
Let me know when you have calculated all the trivial sequences (even powers) up to 120^56 and for the non-trivial (odd powers) up to 120^57.
OK for term size up to 2^256.

[garambois]

Originally Posted by Happy5214

IMO just extend everything beyond base 100 to a fixed digit start value (200 digits maybe?) and leave it at that, whatever exponent that is for that base. That seems round but rigorous enough, though we lose the roundness of the exponent limits themselves.

Thank you for your answer.
We had worked this way at the beginning of the project, but at a 120 digit limit.
It was sometimes laborious, because we always had to go and look at the limit exponent we had to work with.
Especially for yoyo, which reserves bases by large packages.
So I simplified it by putting round numbers in stages for the limit exponents.
And that turned out to be more convenient.
Moreover, for the limit exponents, I tried to find a compromise so that the number of digits in the last sequence of a base is not too big.
I know that no one will calculate sequences of more than 160 digits, at least not until the distant future !
But if other contributors want me to adopt the old method again, I will : all limit exponents for a base calculated so that the first term of the sequence has 200 digits.
Current state of opinion :
- Alexander : limit exponent for each base calculated so that the first term of the last sequence of a base has 200 digits
- Yoyo : leave everything as it is now
- Myself : leave everything as it is now except for b bases such as 280<b<1000 : extend to exponent 70

Originally Posted by Happy5214

That answer seems contradictory. You first say the category is a bad idea, and then you say it could be a good idea? I'm confused. The set of open sequences is pretty steady, especially those below 1e5 (comparing the count on Wikipedia from mid-2015 to today's from the Blue Page, we've only eliminated 7 over that span, and none below 1e4), so while the set is not final, it is stable enough for long-term inclusion.

You are right, the answer seems contradictory !
But I'm really reluctant to create a new category whose bases could then be removed.
But maybe I'm wrong ?
Again, let's make democracy work.
If other contributors give their opinion on the subject, we will make a decision based on the majority that will make it easier for as many people as possible, it seems fair.
Current state of opinion :
- Alexander : add a category including the Open-End sequences of the main project
- Myself : do not add it
On the other hand, if the calculations show that there are really special things happening for these bases, or on the contrary, that nothing is happening at all, this can also make us change our opinion...

Originally Posted by Happy5214

Edit: Unrelated, the credit on 882^44 needs to be changed to me. That was included in https://www.mersenneforum.org/showpo...postcount=1253.

OK, got it.
Sorry, this will be fixed in the next update.

[RichD]

Base 51 can be added at the next update.

[EdH]

Originally Posted by sweety439

. . . 120^3 and 120^5 merges with other sequences . . .

120^3:i0 merges with 1728000:i0
120^5:i77 merges with 442560:i81

[sweety439]

Originally Posted by EdH

120^3:i0 merges with 1728000:i0
120^5:i77 merges with 442560:i81

120^3 = 1728000

[charybdis]

Originally Posted by sweety439

Completed 120^7 at 265 bits ... I will stop only if the number is > 2^256

Okay, maybe I shouldn't be the one to say this as I haven't taken part in this particular sub-project, but surely 256 bits (78 digits) is far lower than everyone else has been going? It took me 6 minutes on one core to reproduce the work you did on 120^7. If it took you much longer than this then there's probably something wrong with your setup.

[VBCurtis]

Welcome to Sweety's idea of contributing work. More time is spent writing posts than computing.

Sweety- You don't need to tell us what happens to each sequence. Just tell us when you're done with the entire base, and the data will all be imported. Cofactors of C100 or terms of 120 digits are more reasonable bounds than 256 bits; Yafu's quadratic sieve is enough firepower for those bounds, so aliqueit can still run without extra interaction on your part. I hope it's not your goal to repeat your CRUS performance with Gary, computing so little that the admin ends up preferring that you hadn't joined at all.