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Old 2014-11-19, 22:54   #12
EdH
 
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"Ed Hall"
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Thanks. I had considered trying to construct some cycles per the described manner, but decided those constructable would have been, to any limit I might easily pursue. Now seeding, I hadn't considered. I may look more into that...
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Old 2014-11-22, 04:07   #13
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Quote:
Originally Posted by Batalov View Post
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.
I started one of my machines looking at the 2E3.11 possibilities, but I had to change the parameters a bit in case the 2E3.11 was in a later part of the cycles than the first (lowest) number. I also increased the permitted cycle size to 9. Not sure how long I'll let this run. Maybe this would be something I can work into my CUDA experiments...
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Old 2014-11-22, 04:38   #14
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Quote:
Originally Posted by EdH View Post
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
A Thank You to Dana Jacobsen (danaj) for pointing me toward his ntheory module for Perl, which has allowed me to discover this cycle much sooner than I would have running the previous PARI version of my search routine.
Very nice find! New sociable numbers are not found very often.

Do there exist sociable numbers of order three? They must be rare, as none have ever been found.
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Old 2014-11-22, 10:49   #15
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Quote:
Originally Posted by philmoore View Post
Very nice find! New sociable numbers are not found very often.

Do there exist sociable numbers of order three? They must be rare, as none have ever been found.
I asked about searching for a 3-cycle here. Borho gave an algrebraic rule for 3-cycles in a 1969 paper which subsequently proved to have an error. In the book referenced in the other thread, the author states they investigated u, v \le 10^{10} using the corrected rule without finding any 3-cycles.

In fact, in the wrap-up for the section on the 3-cycle rule, he said that finding a 3-cycle is likely to be as hard as finding an odd-prefect number, in his opoinion.
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Old 2014-11-22, 16:11   #16
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Thanks. Although I'm limiting my search to 4-cycle (except for the one machine recently changed), they will record any three cycles, if one happens along.
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Old 2014-11-23, 18:15   #17
MichelMarcus
 
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Ed, Congrats for your 4-cycle.
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Old 2014-11-24, 04:53   #18
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Thanks Michel!
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Old 2014-11-24, 15:02   #19
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"Ed Hall"
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Default Wow, another one, already...

I restarted this project in May, after probably a year on and off of previous searching with no success up until recently, and now in less than a week of each other, two have appeared. Odds, anyone?
Code:
16687836720184 = 2^3 * 7 * 79 * 3772114991
19524467203016 = 2^3 * 23 * 439 * 2621 * 92221
18777616502584 = 2^3 * 23 * 113 * 3373 * 267749
18297447537416 = 2^3 * 71 * 179 * 179965453
And, as Batalov may note, this one also has 2E3.71, although it was not found by a focused search.

Thanks to all who have assisted me in my endeavors...
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Old 2014-11-25, 17:38   #20
MichelMarcus
 
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Well done! I can imagine your thrill.
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Old 2014-11-25, 17:54   #21
MichelMarcus
 
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Side notes.
djm.cc/sociable.txt: If you look closely at group 44, you'll see it is still the only group with powers of 2 that are all different; noticed by JM Pedersen when I wrote to him.
OEIS: There are a few sequences related to 4-sociable groups, you can see them by searching for A090615.
Cheers.
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Old 2014-11-26, 04:34   #22
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Quote:
Originally Posted by MichelMarcus View Post
Side notes.
djm.cc/sociable.txt: If you look closely at group 44, you'll see it is still the only group with powers of 2 that are all different; noticed by JM Pedersen when I wrote to him.
OEIS: There are a few sequences related to 4-sociable groups, you can see them by searching for A090615.
Cheers.
Thanks again, Michel,

Am I confusing something or did you mean powers of 3? I see they are 5 4 2 3. That is interesting.

Thanks also for the OEIS link as well.
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