Sociable Numbers

[EdH]

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 5x10¹², but that's of a few years ago.

Is anyone actively seeking them now?

[Drdmitry]

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.

[EdH]

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 four-order 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.

[EdH]

Originally Posted by Drdmitry

...
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.

Well, I've refined(?) a pari routine that seems to run well, but I'm wondering if the numbers you omitted in your search were because sociables would not exist, or the time involved would be outside your workable zone.

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.

[Drdmitry]

Originally Posted by EdH

Well, I've refined(?) a pari routine that seems to run well, but I'm wondering if the numbers you omitted in your search were because sociables would not exist, or the time involved would be outside your workable zone.

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.

Well, the method I used works much faster if the Aliquot sequence goes down on average. So it was most optimal for odd numbers, however it also worked quite well for downdrivers. I thought about continuing the search for numbers starting with 2^2 times a prime number bigger than 7 but then I was distracted by other things.
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.

[EdH]

Originally Posted by Drdmitry

Well, the method I used works much faster if the Aliquot sequence goes down on average. So it was most optimal for odd numbers, however it also worked quite well for downdrivers. I thought about continuing the search for numbers starting with 2^2 times a prime number bigger than 7 but then I was distracted by other things.
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.

Thanks. I'm turning up amicables as a side-product on my way through four iterations. If I'm correct in my programming, which is intended to skip what you have searched, then I will be checking all the numbers you omitted. My current program checks only 0 mod 4 and even 0 mod 3 initial numbers and a test found 430324482433184, which is from the list of known sociables. My machines are currently spanning 5.5-5.7*10^12.

Thanks again for the info.

[EdH]

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...)

[MichelMarcus]

Cycle number 56 was discovered by Andre Needham.
See this line in http://djm.cc/sociable.txt
42,46,48-51,53-57: A. Needham

[EdH]

Originally Posted by MichelMarcus

Cycle number 56 was discovered by Andre Needham.
See this line in http://djm.cc/sociable.txt
42,46,48-51,53-57: A. Needham

Thanks - I overlooked it somehow and a search for "56" was obviously ineffective...

[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:

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.

[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.