20190815, 11:22  #23 
Mar 2015
Australia
82_{10} Posts 
I've mentioned to Garambois via email that I've tried looking for small cycles of length 3 or more without success.
However, I had been using pari's factor() function in only the normal mode,so normal numbers would get a normal value of the sigma function. Newer versions of pari allow this: ? factor(8,I) %16 = [ I 1] [1 + I 6] Giving a complex factorisation and a different value for sigma I'm planning to do some searching with this over smaller ranges, Andrew 
20190815, 12:39  #24 
Oct 2011
2^{4}×3×5 Posts 
Thank you Andrew for your help via email.
I finally managed to write successfully a program for the Sage software that runs and calculates sigma(z) for z a complex integer Gauss number. I am looking for cycles of length c with 1<=c<=50 for all complex numbers z=a+b*I for 0<=a<=3000 and 0<=b<=3000 (a and b integers of course). But it may take several years ! The program is extremely slow, because the terms of the complex aliquot sequence sometimes become difficult to factor after 40 iterations ! 
20190816, 07:59  #25 
Mar 2015
Australia
2·41 Posts 
I'm changing a few pari scripts I wrote a while back and will give them a run. These will
mainly be randomised searches One suggestion for anyone looking at these. The complex pairs I've found so far have all had (1+I) , (1+2*I) or (1+4*I) as a factor. So a search over a defined range (or a random search) could also work by multiplying the value by one of these. Possibly to a small power. For two cycles this will greatly improve the chance of success, for longer cycles??? garambois I'd also suggest running on a separate processor/computer a search just up to 4 or even 6 length as it will get to larger values much quicker. If you do a 50 cycle search, think carefully of what upper limit you will let these reach as some values might increase rapidly just as in real numbers! Andrew Last fiddled with by AndrewWalker on 20190816 at 08:00 
20190816, 09:11  #26  
Oct 2011
F0_{16} Posts 
Quote:
Unfortunately my computer only has 12 threads. And I have many more ideas to test. I also want to calculate aliquot sequences starting on integer powers. Maybe in a while.... I dream of a 120 thread computer ! 

20190820, 11:15  #27 
Oct 2011
240_{10} Posts 
Does anyone know if there exist Gauss untouchable numbers, with the sigma(z)z function, for z a complex number, by analogy to the ErdÃ¶s untouchable numbers with the sigma(n)n function, with n an integer (https://oeis.org/A005114) ?
Of course, in the context of this question, for the calculation of sigma(z)z, we consider the factorization of z with gaussian prime factors only taken in the first quadrant. 
20191022, 12:51  #28 
Oct 2011
2^{4}×3×5 Posts 
Paul ZIMMERMANN discovered a Gaussian Aliquot Cycle of length 6 !
507253 + 70523 * I Complete Gauss 6_cycle : 0 70523*I  507253 1 335537*I  225727 2 373843*I + 112387 3 130729*I + 243117 4 433407*I  155165 5 294287*I + 139123 Congrats to Paul ! Can anyone confirm it, in addition to Paul and myself ? To my knowledge, this is the first known Gaussian Aliquot Cycle of a length other than 2, isn't it ! I had told Paul about my research program for such cycles and I told him how slow it was. He then decided to write a program in C much faster. Here is the result after a few hours of operation of this new program ! Last fiddled with by garambois on 20191022 at 13:46 
20191023, 06:51  #29 
Mar 2015
Australia
2·41 Posts 
Huge congratulations to Paul, amazed he did this so quick! I hope his code is much
quicker than what I've been doing in pari as it could then be quite useful! Have confirmed it in pari, CSigma function is earlier in this thread: (17:36) gp > a= 70523*I  507253 %2 = 507253 + 70523*I (17:37) gp > b=CSigma(a)a %3 = 225727 + 335537*I (17:37) gp > c=CSigma(b)b %4 = 112387 + 373843*I (17:37) gp > d=CSigma(c)c %5 = 243117 + 130729*I (17:39) gp > e=CSigma(d)d %6 = 155165 + 433407*I (17:39) gp > f=CSigma(e)e %7 = 139123 + 294287*I (17:40) gp > CSigma(f)fa %8 = 0 (17:40) gp > factor(a) %9 = [ 1 + I 1] [ 21 + 26*I 1] [ 18 + 35*I 1] [268 + 63*I 1] (17:46) gp > factor(b) %10 = [ 1 + I 1] [ 62 + 3*I 1] [ 4 + 5*I 1] [653 + 302*I 1] (17:46) gp > factor(c) %11 = [ 1 + I 1] [243115 + 130728*I 1] (17:46) gp > factor(d) %12 = [ I 1] [ 1 + I 1] [ 1 + 2*I 1] [ 29 + 4*I 1] [2980 + 103*I 1] (17:46) gp > factor(e) %13 = [ 1 + I 1] [139121 + 294286*I 1] (17:47) gp > factor(f) %14 = [ I 1] [ 1 + I 1] [ 19 + 50*I 1] [3272 + 2795*I 1] (17:47) gp > Again very nice hope this will result in many more to follow! Maybe (1+I) * 1 or 2 gaussian primes is a good form to search? (possibly with I as the unit.) Are 1 and +I possible or not? Today I finally reached 900 pairs in my 2 cycle search, will do an updated list by or on the weekend. Andrew 
20191023, 08:31  #30  
Oct 2011
360_{8} Posts 
Quote:
We have used your lists for our ongoing work... Thank you for the confirmation of the discovery of the cycle of length 6 ! 

20191028, 07:12  #31 
Mar 2015
Australia
2×41 Posts 
Here's the latest update, now 910 pairs! One pair from the Ranthony Ashley Clark thesis,
a (5,5) type pair had eluded me but I managed to find it earlier in the year by tweaking one of my searches and going deeper on it than some others. Will follow with a version showing factors. Andrew 
20191028, 08:22  #32 
Mar 2015
Australia
2×41 Posts 
List with factors included, hopefully haven't goofed up, let me know if I have!
Andrew 
20191102, 18:42  #33 
Oct 2011
F0_{16} Posts 
I just got back from vacation.
What a pleasant surprise ! Thank you very much for these new lists. They will replace the old ones in our programs... Paul Zimmermann's program is running right now. No other cycle of length other than 2 until now, which would be added to the one of length 6 announced above. 
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