20200708, 12:51  #331  
Oct 2011
2×131 Posts 
Quote:
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 Code:
[[2, 13], [3, 4], [7,4], [11,1], [19, 1], [31, 1], [41, 1]] 

20200708, 15:32  #332  
"Ed Hall"
Dec 2009
Adirondack Mtns
2^{4}×7×29 Posts 
Quote:
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 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 20200709 at 16:08 

20200708, 19:42  #333 
Oct 2011
262_{10} Posts 
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. I think for the highest peak, the term is better, because there's more information than if you just give the size. 
20200708, 20:05  #334 
Oct 2011
2×131 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 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 
20200708, 21:29  #335  
"Ed Hall"
Dec 2009
Adirondack Mtns
3248_{10} Posts 
Quote:
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 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? 

20200709, 06:57  #336 
Oct 2011
262_{10} Posts 

20200709, 07:26  #337 
Oct 2011
2×131 Posts 
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 ! Last fiddled with by garambois on 20200709 at 07:33 
20200709, 07:59  #338 
Mar 2006
Germany
2×3×11×43 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 subgraphs 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 
20200709, 16:06  #339 
"Ed Hall"
Dec 2009
Adirondack Mtns
2^{4}·7·29 Posts 
Here are corrected statistics for bases 27:

20200709, 16:20  #340 
Sep 2008
Kansas
2·3^{2}·167 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 parttime.

20200709, 16:29  #341  
"Ed Hall"
Dec 2009
Adirondack Mtns
2^{4}×7×29 Posts 
Quote:
Quote:
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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