mersenneforum.org Aliquot sequences that start on the integer powers n^i
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2020-07-08, 12:51   #331
garambois

Oct 2011

35010 Posts

Quote:
 Originally Posted by EdH For later, as to primes found within a sequence, will you need a count of each or just a list?
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 :

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

2020-07-08, 15:32   #332
EdH

"Ed Hall"
Dec 2009

1101001100112 Posts

Quote:
 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 : Code: 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 : Code: [[2, 13], [3, 4], [7,4], [11,1], [19, 1], [31, 1], [41, 1]]
I will work on this, but it might be a little bit before I have it complete.

Meanwhile, I am attaching bases 2 - 7 statistics with the new format. I added 2 (prime) and 3 (cycle) to the c position, since they are neither abundant nor deficient. I also skipped all the open sequences. I used the following key, with no title, in the documents:
Code:
[a, b, c, d, e, f, g, h, i, j, k, l, m, n]
a : base
b : exponent
c : 0 if deficient, 1 if abundant, 2 if prime, 3 if perfect
d : number of total abundant terms
e : number of total deficient terms
f : most sequential abundant terms
g : most sequential deficient terms
h : parity changes even to odd
i : parity changes odd to even
j : number of abundant peaks
k : greatest abundant peak (2dd)
l : greatest peak larger than first term 1 if yes and 0 if no
m : last index
n : last term
I hope I got all this correct!

Edit: Actually, did you want the size of the highest peak or the term? I provided the term, but I can change that and rerun easily.

Edit2: The attachments were removed because they were flawed. The new (current) files can be found in this new post.

Last fiddled with by EdH on 2020-07-09 at 16:08

2020-07-08, 19:42   #333
garambois

Oct 2011

2×52×7 Posts

Quote:
 Originally Posted by EdH I hope I got all this correct!
I'll check it out, maybe even try to replicate your algorithm.
But I'm not sure I'll have the time to do all this before I go on holiday in a few days : there's nothing you can do against the call of the mountain !
So far I've done some manual checks, everything seems to be fine. But it's laborious to do so...

Before I leave, I think I'll concentrate on counting all the prime numbers of all the terms in a sequence, as in the example above.

In August, when I come back, I will have much more time for everything else.

Quote:
 Originally Posted by EdH Edit: Actually, did you want the size of the highest peak or the term? I provided the term, but I can change that and rerun easily.
I think for the highest peak, the term is better, because there's more information than if you just give the size.

 2020-07-08, 20:05 #334 garambois     Oct 2011 2·52·7 Posts I have two questions: 1) Please, I'm not sure I understand the meaning of : Code: l: greatest peak larger than first term 1 if yes and 0 if no Why, in the case of 2^i, is it always worth 1 ? Indeed, the starting term of the sequence is always the largest, isn't it ? 2) The value of c is only about the first term of the sequence, isn't it ? Code: c : 0 if deficient, 1 if abundant, 2 if prime, 3 if perfect
2020-07-08, 21:29   #335
EdH

"Ed Hall"
Dec 2009

D3316 Posts

Quote:
 Originally Posted by garambois I have two questions: 1) Please, I'm not sure I understand the meaning of : Code: l: greatest peak larger than first term 1 if yes and 0 if no Why, in the case of 2^i, is it always worth 1 ? Indeed, the starting term of the sequence is always the largest, isn't it ? 2) The value of c is only about the first term of the sequence, isn't it ? Code: c : 0 if deficient, 1 if abundant, 2 if prime, 3 if perfect
1) I have something wrong for that value. It is not always 1, but there are 60 sequences (2^1 - 2^539) where a peak exceeds the initial term. For example, see 2^12:
Code:
0 .   4096 = 2^12
1 .   4095 = 3^2 * 5 * 7 * 13
2 .   4641 = 3 * 7 * 13 * 17
3 .   3423 = 3 * 7 * 163
4 .   1825 = 5^2 * 73
5 .   469 = 7 * 67
6 .   75 = 3 * 5^2
7 .   49 = 7^2
8 .   8 = 2^3
9 .   7 = 7
Term 2 (4641) > term 0 (4096). Those 60 should show 1. I will have to fix that, but it will have to wait until later tonight or tomorrow morning. Sorry for the delay.

2) That is correct. The value c is about term 0 only. If term 0 is perfect, s(n)-n is equal to itself (neither deficient nor abundant), if prime, it's equal to 1 (deficient, but unique). I could flag primes as deficient, but it really wouldn't be the same as the others, since there are no terms after a prime termination. How would you like me to reflect perfect and prime for your needs?

2020-07-09, 06:57   #336
garambois

Oct 2011

2×52×7 Posts

Quote:
 Originally Posted by EdH 1)Term 2 (4641) > term 0 (4096). Those 60 should show 1. I will have to fix that, but it will have to wait until later tonight or tomorrow morning. Sorry for the delay.

As I said before, I'll be working on this in August...

2020-07-09, 07:26   #337
garambois

Oct 2011

1010111102 Posts

The first array is finished.

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 !
Attached Files
 Aliquot sequences n^i prime or end cycle numbers.pdf (43.9 KB, 10 views)

Last fiddled with by garambois on 2020-07-09 at 07:33

2020-07-09, 07:59   #338
kar_bon

Mar 2006
Germany

285010 Posts

2 years ago I played around with aliquot sequences and graphviz, a free tool to display graphs.

The problem is you can not only include all relations automatically because the graph will grow and isn't easy to handle. So you have to create some sub-graphs to tidy up the image.

I've created a tree of squences ending in 43 with (I think all values<20000, no odd) and the result is an image.
The source is a text file ~10kB in the graphviz syntax (the extension has to be changed into 'gv' for graphviz).
The image was created with
Code:
dot -Tjpg 43.gv >43.jpg
The problem here: the filesize is ~1.8MB (too big for jpg) and the dimensions (~10,000 x 2900 pixel, to big, too), so I uploaded it into the Wiki.
Attached Files
 43.txt (8.9 KB, 8 views)

2020-07-09, 16:06   #339
EdH

"Ed Hall"
Dec 2009

31·109 Posts

Here are corrected statistics for bases 2-7:
Attached Files
 2-Stats.txt (45.9 KB, 6 views) 3-Stats.txt (10.1 KB, 6 views) 5-Stats.txt (6.6 KB, 7 views) 6-Stats.txt (10.6 KB, 6 views) 7-Stats.txt (6.1 KB, 6 views)

 2020-07-09, 16:20 #340 RichD     Sep 2008 Kansas 13×241 Posts I've started base 30 for n=1 to 20. At least half have trivially terminated. It will be slow going because I am using a C2D laptop part-time.
2020-07-09, 16:29   #341
EdH

"Ed Hall"
Dec 2009

D3316 Posts

Quote:
 Originally Posted by kar_bon 2 years ago I played around with aliquot sequences and graphviz, a free tool to display graphs. . . .
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?

Quote:
 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 : Code: 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 : Code: [[2, 13], [3, 4], [7,4], [11,1], [19, 1], [31, 1], [41, 1]]
Do you need the sequence referenced or will the list suffice as each line corresponding to that power?

The table shows some interesting values, but as you say, these base tables are such a tiny portion of the overall, it's tough to find any commonalities.

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