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-   Aliquot Sequences (https://www.mersenneforum.org/forumdisplay.php?f=90)

 schickel 2012-03-24 01:51

 Dubslow 2012-03-24 02:02

[QUOTE=schickel;293973]....in the meantime I redirected the Analysis link to The Wayback Machine, but I wonder if we should rehost the info he had posted.[/QUOTE]
That's a great idea, I can't believe I didn't think of that. And thanks for stooping to my math level; unfortunately, discrete math/number theory has never really appealed to me (still doesn't, to be honest), but GIMPS is such an awesome DC project. Now that I'm here though, it seems interesting, but I need to fill in the mathematics :razz: Thanks for helping.

(Among other deficiencies is a complete lack of knowledge of factoring methods besides ECM, of which I only have a basic idea that it's similar to P-1 and uses elliptic curves. I've gleaned that the first line of attack is ECM, but I'm not sure when to switch methods or what to use. You got any more links? :P)

 Bode Didymos 2016-01-16 23:33

About the starting value of an Aliquot Sequence

I have question about the starting value of a aliquot sequence. OP said that an Aliquot sequences are generally referred to by their starting value, is there some numbers that start an Aliquot Sequence but is never in the middle of another aliquot sequence? how do you call those numbers? these numbers would be those that are not in the image of the aliquot sum function. Another related question, if such "patriarch numbers" exist (or what ever you call them), does every branch of an aliquot family tree have a "patriarch" that initiate that branch or its goes on and on indefinitely?

Thank you for your time =D

 Happy5214 2016-01-17 10:45

I will answer the first part of your question and [I]try[/I] to come up with something for the second part. An [I]untouchable number[/I] is a number that does not occur as the aliquot sum of any other number. There are infinitely many untouchable numbers, it is conjectured that only one is odd (5), and it is also believed that all but 2 and 5 are composite. [URL="https://oeis.org/A005114"]This[/URL] is a list of untouchable numbers below 700.

The second part is a little trickier. I would imagine that every full sequence branches from an untouchable number. (Could someone more knowledgeable confirm that?) But don't confuse that untouchable number with the starting value we use. We basically refer to sequences by their lowest value. For example, 564 is used as a starting value, but it is [I]not[/I] an untouchable number as it is the aliquot sum of 563^2. Also, it is conjectured, but not yet proven, that all sequences terminate with a prime, perfect number, or aliquot cycle. There [I]could[/I] be infinitely long sequences that never terminate. So that answer to both parts of your second question could be "yes."

 henryzz 2016-01-18 13:58

[QUOTE=Happy5214;422779]I will answer the first part of your question and [I]try[/I] to come up with something for the second part. An [I]untouchable number[/I] is a number that does not occur as the aliquot sum of any other number. There are infinitely many untouchable numbers, it is conjectured that only one is odd (5), and it is also believed that all but 2 and 5 are composite. [URL="https://oeis.org/A005114"]This[/URL] is a list of untouchable numbers below 700.

The second part is a little trickier. I would imagine that every full sequence branches from an untouchable number. (Could someone more knowledgeable confirm that?) But don't confuse that untouchable number with the starting value we use. We basically refer to sequences by their lowest value. For example, 564 is used as a starting value, but it is [I]not[/I] an untouchable number as it is the aliquot sum of 563^2. Also, it is conjectured, but not yet proven, that all sequences terminate with a prime, perfect number, or aliquot cycle. There [I]could[/I] be infinitely long sequences that never terminate. So that answer to both parts of your second question could be "yes."[/QUOTE]
I think that stating that every sequence starts at an untouchable number might be similar to stating that all sequences terminate as you could just as easily have an infinite sequence backward as forwards.
I would guess that it would be much less likely to happen as numbers in general get bigger as you go upward in a sequence and smaller as you go down. Numbers are limited in how much they can go down so it is less likely to happen.
We do get long sequences reaching smaller numbers than their starting value(i.e. merging with a smaller sequence).
Need to get on with work now. Might think more later.

 bur 2021-07-23 07:14

I noticed that some open end sequences (e.g. 26236) aren't in the blue page reservation table. Are those sequences that merge with others? I couldn't find this number in the terminations/mergers thread though.

 Drdmitry 2021-07-23 08:46

[QUOTE=bur;583810]I noticed that some open end sequences (e.g. 26236) aren't in the blue page reservation table. Are those sequences that merge with others? I couldn't find this number in the terminations/mergers thread though.[/QUOTE]

You can check that 26236:i3 coincides with 4800:i7.Therefore the open sequence which started from 26236 is the same as 4800.

 bur 2021-07-23 10:29

Where did you check it? I used the forum search and it came up empty.

 Drdmitry 2021-07-27 09:47

[QUOTE=bur;583817]Where did you check it? I used the forum search and it came up empty.[/QUOTE]

I would start with looking at the last term of sequence 26236 on [URL="http://factordb.com/sequences.php?se=1&aq=26236&action=last20&fr=0&to=100"]factordb[/URL]. Then I would go to the [URL="https://www.rechenkraft.net/aliquot/AllSeq.html"]blue page[/URL] and look for the sequence which ends in the same factorisation. We know that it should start with something smaller than 26236, which simplifies the task.

 EdH 2021-07-27 12:51

[QUOTE=bur;583817]Where did you check it? I used the forum search and it came up empty.[/QUOTE]Check out [URL="https://www.mersenneforum.org/showthread.php?t=24423&page=3"]this thread[/URL]. The "margins" is really "merges."

Also, I thought you were already running my alimerge3 program.: [URL]https://www.mersenneforum.org/showpost.php?p=582143&postcount=1201[/URL][code]\$ ./alimerge3 26236 1 1
Running base 26236 from 1 through 1 . . .
26236^1:i3 merges with 4800:i7
Run took 15 seconds.[/code]

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