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 2014-11-19, 22:54 #12 EdH     "Ed Hall" Dec 2009 Adirondack Mtns 3,253 Posts 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...
2014-11-22, 04:07   #13
EdH

"Ed Hall"
Dec 2009

1100101101012 Posts

Quote:
 Originally Posted by Batalov 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...

2014-11-22, 04:38   #14
philmoore

"Phil"
Sep 2002
Tracktown, U.S.A.

21358 Posts

Quote:
 Originally Posted by EdH 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.

Do there exist sociable numbers of order three? They must be rare, as none have ever been found.

2014-11-22, 10:49   #15
schickel

"Frank <^>"
Dec 2004
CDP Janesville

84A16 Posts

Quote:
 Originally Posted by philmoore 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.

 2014-11-22, 16:11 #16 EdH     "Ed Hall" Dec 2009 Adirondack Mtns 325310 Posts 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.
 2014-11-23, 18:15 #17 MichelMarcus   Nov 2011 Saint Maur, France 628 Posts Ed, Congrats for your 4-cycle.
 2014-11-24, 04:53 #18 EdH     "Ed Hall" Dec 2009 Adirondack Mtns 3,253 Posts Thanks Michel!
 2014-11-24, 15:02 #19 EdH     "Ed Hall" Dec 2009 Adirondack Mtns 3,253 Posts 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...
 2014-11-25, 17:38 #20 MichelMarcus   Nov 2011 Saint Maur, France 2·52 Posts Well done! I can imagine your thrill.
 2014-11-25, 17:54 #21 MichelMarcus   Nov 2011 Saint Maur, France 2×52 Posts 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.
2014-11-26, 04:34   #22
EdH

"Ed Hall"
Dec 2009

3,253 Posts

Quote:
 Originally Posted by MichelMarcus 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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