Aliquot sequences that start on the integer powers n^i

[garambois]

@RichD :

OK, many thanks. I will add base 30 in the next update, in 2 or 3 days.



@Kar_bon :

Yes, this is impressive ! thank you for the link.



@EdH :

To answer your questions :

1) Yes, I think 43 appears to be the most common termination at higher numbered sequences >1M. On my website, I have a database called "fundamental database." But sorry, all explanations are in french.
Thanks to this base, I was able to determine that among all the sequences that start with the integers from 1 to 10M, there are exactly 666638 that end with the prime number 43! There are 456843 that end with 59 and 437318 that end with 41. See here.
But to build this database, I considered the sequences to be open-end as soon as the size of the terms exceeded 50 digits. So there are even more that must end with those prime numbers...

2) I don't need the sequence referenced. The list is enough because each line corresponds to the power.



And a more general comment on the project :

Frankly, I wonder whether it wouldn't make sense to push all the bases up to 160 digits, but only for sequences that end trivially on a prime number. I'm still not sure whether pushing the calculations that far would bring an interesting gain in terms of improving the statistics ?
Or is it more interesting to calculate other bases up to 120 digits ?

[kar_bon]

Originally Posted by EdH

This is impressive! I remember some of it from much longer ago, when the genealogy thread was active. Does 43 appear to be the most common termination at higher numbered sequences (>1M), as well?

On Wolfgang Creyaufmuellers page you can find a table with endings for all seqs <1M (eventually outdated) (download as zip also available, see on top under the first table "About 1% of all integers are...").

Here the first 5 most families:

            43     78060        7.81%
            59     53197        5.32%
            41     51012        5.10%
             7     42299        4.23%
           601     26759        2.68%

[EdH]

@garambrois: Does this mean all the sequences <10M are represented in factordb at >= 50 dd? I believe they would be updated gradually by the "elves" only when accessed, but any access would move their last terms. (I might verify that later.)

It would be easier by far to add more bases than to extend to 160 dd. I'm taking roughly 1.5 days on average to factor 15x dd in the base 2 table, with my whole "farm." I think smaller equipped contributors would be less likely to participate.

@kar_bon: Thanks for the links. I haven't been to his page(s) in quite some time. I'll have to spend some more time there.

[EdH]

Originally Posted by garambois

Thanks, Ed.
I'll need a count of every prime number, the list alone is not enough !
For a given sequence, how many times have we had the two, three, five, seven...
For example, for the sequence starting at 2^10, that would be :

Sequence :

0 .   1024 = 2^10
1 .   1023 = 3 * 11 * 31
2 .   513 = 3^3 * 19
3 .   287 = 7 * 41
4 .   49 = 7^2
5 .   8 = 2^3
6 .   7 = 7

Result :

[[2, 13], [3, 4], [7,4], [11,1], [19, 1], [31, 1], [41, 1]]

I have a script to do what you've shown above:
2^10 matches yours (except for order*):

[[2, 13], [3, 4], [7, 4], [11, 1], [31, 1], [19, 1], [41, 1]]

and, 5^4 shows the following:

 [[2, 13], [3, 1], [5, 4], [7, 3], [13, 1], [59, 1], [23, 1], [11, 1], [29, 1], [37, 1]]

*I haven't provided a sort. Is that necessary?

Now, something I'm concerned about: the practicality of capturing every prime. I ran 5^6 and came up with a single line that was well over 225000 characters. Should I possibly trim the primes in some manner? A full single table will be enormous.

[garambois]

Originally Posted by EdH

@garambrois: Does this mean all the sequences <10M are represented in factordb at >= 50 dd? I believe they would be updated gradually by the "elves" only when accessed, but any access would move their last terms. (I might verify that later.)

It would be easier by far to add more bases than to extend to 160 dd. I'm taking roughly 1.5 days on average to factor 15x dd in the base 2 table, with my whole "farm." I think smaller equipped contributors would be less likely to participate.

1) No, I never entered this data into factordb. But calculating all open-ends up to 10M up to 50 digits should only take a few hours or days...
What takes time in building up my database is the fact that the side tables keep growing (regina_cycles, regina_opens and regina-prems, on http://www.aliquotes.com/aliquote_base.htm#alibasefonda).
Calculations have been in progress for more than 2 years and I'm at 13.5M.

2) OK, I'll take your advice and calculate other bases instead of going up to 160 figures for all the bases. I'll be doing this job in two years, when I have a 64C/128T CPU !

[garambois]

Originally Posted by EdH

I have a script to do what you've shown above:
2^10 matches yours (except for order*):

[[2, 13], [3, 4], [7, 4], [11, 1], [31, 1], [19, 1], [41, 1]]

and, 5^4 shows the following:

 [[2, 13], [3, 1], [5, 4], [7, 3], [13, 1], [59, 1], [23, 1], [11, 1], [29, 1], [37, 1]]

*I haven't provided a sort. Is that necessary?

Now, something I'm concerned about: the practicality of capturing every prime. I ran 5^6 and came up with a single line that was well over 225000 characters. Should I possibly trim the primes in some manner? A full single table will be enormous.

1) It is best if you sort the prime numbers in ascending order in the table. But maybe that's too much work ? No problem if it's too much work, I'll be able to program the sorting myself, because I can imagine how much time you spend to write all these algorithms, and really, a big thank you for all this time you give !!!

2) Yes, the tables can be huge !
But it's still interesting to include all the prime numbers, even if at first, I will only use prime numbers below 1000. Sometimes we have surprises. For example, for all the integers from 1 to 10M, there are 69 ending with the prime 4737865361 or 5 ending with the prime 14604141802777 (see here). These large prime numbers are sequence "attractors", such as 43.
The question is to know if there are such attractors of any size in the set of all sequences and, as far as our project is concerned, in the set of all sequences starting on numbers composed of only 2, only 3, only 5, only 7...

[EdH]

2) This does cause further questions I had not considered earlier:

a - Does a large termination prime appear earlier as a factor?
b - Do termination primes appear in other sequences in an interesting frequency?

The answers would only be available if all primes are included in the lists.

One thing I've often thought would be a nice feature for factordb, would be all the sequences associated with a single prime. It would be nice if that were available in the "Info" section, but it would be prohibitively large for smaller primes.

[Happy5214]

I'm done with 21^62 and 21^64 and will release those.

I'll just go ahead and reserve the rest of base 21 to i=97.

Originally Posted by EdH

It would be easier by far to add more bases than to extend to 160 dd. I'm taking roughly 1.5 days on average to factor 15x dd in the base 2 table, with my whole "farm." I think smaller equipped contributors would be less likely to participate.

I concur wholeheartedly. My lowly computer can't do 160 digits. In fact, I'm sitting on a 5-digit downdriver sequence because I don't want to do the C144 cofactor (ECMed to ~t48) until I get newer/better hardware.

[garambois]

Originally Posted by EdH

2) This does cause further questions I had not considered earlier:

a - Does a large termination prime appear earlier as a factor?
b - Do termination primes appear in other sequences in an interesting frequency?

The answers would only be available if all primes are included in the lists.

One thing I've often thought would be a nice feature for factordb, would be all the sequences associated with a single prime. It would be nice if that were available in the "Info" section, but it would be prohibitively large for smaller primes.

a) No, in my opinion, we don't need to distinguish between prime numbers at the end of sequences and others here. This study is done with the other program.
I just finished writing the program for all the primes in a sequence. I'm running it for Base 2. I'm taking into account all the sequences, even the open-ends. I'll communicate the results for a few bases in a few hours or tomorrow depending on how long the calculations take...

b) I never thought of that !
It's an interesting question, it's a good idea : I'll think about it...

Knowing all the sequences associated with a prime number would be interesting, but it must be difficult to program and very time-consuming in terms of calculation time and memory. I remember having done some tests on this. But maybe I didn't do it right !

[garambois]

Originally Posted by Happy5214

I'm done with 21^62 and 21^64 and will release those.

I'll just go ahead and reserve the rest of base 21 to i=97.

I concur wholeheartedly. My lowly computer can't do 160 digits. In fact, I'm sitting on a 5-digit downdriver sequence because I don't want to do the C144 cofactor (ECMed to ~t48) until I get newer/better hardware.

Ok, thanks a lot.
I change that in tomorrow's update...

Yes, on reflection, I also agree with you that our computers are not yet able to do the work up to 160 figures in a reasonable time ...

[EdH]

Originally Posted by garambois

. . .
I just finished writing the program for all the primes in a sequence. I'm running it for Base 2. I'm taking into account all the sequences, even the open-ends. I'll communicate the results for a few bases in a few hours or tomorrow depending on how long the calculations take...
. . .

I, too, have a working program/script that now includes sorting. But, if you already have something working, I won't bother uploading files that may be too large anyway.

Are you still interested in the earlier abundancy/deficiency, etc. counts for the tables above the five I already provided? I have tables up through base 28 done and can continue with the higher ones.