20140518, 03:53  #1 
"Ed Hall"
Dec 2009
Adirondack Mtns
2·1,607 Posts 
Sociable Numbers
Everything I'm finding on line is several years old. Where would I locate anything current? One question is what the exhaustive search limit has reached for all orders. For order 4, it seems to be 5x10^{12}, but that's of a few years ago.
Is anyone actively seeking them now? 
20140521, 09:37  #2 
Nov 2011
233 Posts 
The most recent list of known sociable cycles can be found here. The last time it was updated on March 2014.
About four years ago I did the search of all cycles such that the minimal element is odd and is <=10^14; the maximal element is <=10^16 and the length is <=200. About two years ago I extended the search to the cycles with minimal element divisible by 2 but not divisible neither by 4 nor by 3. I do not think that anyone did the full search of sociable numbers beyond 5*10^12. 
20140521, 13:11  #3 
"Ed Hall"
Dec 2009
Adirondack Mtns
2·1,607 Posts 
Thanks for the reply. As you may remember, I have "played around" with these before and am dabbling again. My previous ventures were in the 10^15 area, but quite spotty.
I had been reviewing the list you provided, but thank you for referencing it, in case I wasn't aware. Although the date shows 2014, the latest addition to the list seems to be from a long time previous. I guess I can consider this list complete, since the date is current? I have renewed my interest and am playing with some more pari scripts, but have narrowed the parameters. I'm currently running a fourorder only search right at 5x10^12, but am searching both odd and even. Since you've covered the odd pattern, I will see if I can gain improvement by narrowing my tests to even elements. Not sure how long my attention span will extend this time, but for now, I'll run a "few" checks and see if I can rediscover the already known cycles. 
20140605, 15:15  #4  
"Ed Hall"
Dec 2009
Adirondack Mtns
2·1,607 Posts 
Quote:
I've set up my machines to cover the numbers above 5*10^12 that you skipped, if I understood your description correctly, but I'm only set up to catch amicables and order four sequences. Am I wasting my time in this region? If so, is there a better region and search criteria? Thanks. 

20140606, 09:59  #5  
Nov 2011
233 Posts 
Quote:
One can modify the algorithm so it becomes more efficient not only for decaying sequences by starting with the number prior the largest number in the cycle. (This trick is explained in D. Moews, P. Moews 1993 paper). Aliquot cycles containing drivers are theoretically possible but they are very unlikely (especially in such a "small" range as 5*10^12  10^15). So I did not check them. Finally it is useless to look for the amicable pairs in the range 5^10^12  10^15 since the complete search of them has already been done (I do not remember by whom). On the other hand it makes sense to look for the cycles of length four. I guess there should be 2  4 of them not discovered yet. 

20140606, 14:25  #6  
"Ed Hall"
Dec 2009
Adirondack Mtns
2×1,607 Posts 
Quote:
Thanks again for the info. 

20140814, 01:55  #7 
"Ed Hall"
Dec 2009
Adirondack Mtns
110010001110_{2} Posts 
After a long break (of a couple months), I have renewed my efforts and restarted many of my machines.
@Drdmitry: On Moews' page "A LIST OF ALIQUOT CYCLES OF LENGTH GREATER THAN 2" I can't find the credit for cycle number 56. Am I missing something right in front of me? More to my real question, did you discover this one, or do you know how it was found? This cycle is the only one in the list which will meet my search criteria any time relatively soon and I hope it proves my script valid when I reach it. (Actually, I hope even more to find a new one...) 
20140922, 15:02  #8 
Nov 2011
Saint Maur, France
2×5^{2} Posts 
Cycle number 56 was discovered by Andre Needham.
See this line in http://djm.cc/sociable.txt 42,46,4851,5357: A. Needham 
20140923, 13:27  #9  
"Ed Hall"
Dec 2009
Adirondack Mtns
2×1,607 Posts 
Quote:


20141119, 19:27  #10 
"Ed Hall"
Dec 2009
Adirondack Mtns
2·1,607 Posts 
I have found one  an aliquot cycle of order four  that I have not found listed anywhere, as of yet. Is it possible this is a new discovery? The cycle is:
Code:
14592614233912 = 2^3 * 11 * 13 * 263 * 48501071 17674178946248 = 2^3 * 11 * 2417 * 8783 * 9461 18500448943672 = 2^3 * 71 * 60527 * 538127 16677107567048 = 2^3 * 220511 * 9453671 
20141119, 19:31  #11 
"Serge"
Mar 2008
Phi(3,3^1118781+1)/3
5·1,811 Posts 
Nice one! Congrats!
Looking at djm.cc list, "2E3.11." and "2E3.71." might be good seeds for a focused search for similar ones. EDIT: Is it of a form given by Borho ? No, it is not. Last fiddled with by Batalov on 20141119 at 19:56 
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