20120324, 01:51  #12  
"Frank <^>"
Dec 2004
CDP Janesville
2·1,061 Posts 
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20120324, 02:02  #13  
Basketry That Evening!
"Bunslow the Bold"
Jun 2011
40<A<43 89<O<88
3·29·83 Posts 
Quote:
(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 P1 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) 

20160116, 23:33  #14 
Jan 2016
1 Posts 
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 
20160117, 10:45  #15 
"Alexander"
Nov 2008
The Alamo City
11×29 Posts 
I will answer the first part of your question and try to come up with something for the second part. An untouchable number 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. This 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 not 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 could be infinitely long sequences that never terminate. So that answer to both parts of your second question could be "yes." 
20160118, 13:58  #16  
Just call me Henry
"David"
Sep 2007
Cambridge (GMT)
2·2,837 Posts 
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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. Last fiddled with by henryzz on 20160118 at 13:58 

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