20210324, 21:24  #1035 
Just call me Henry
"David"
Sep 2007
Cambridge (GMT/BST)
13356_{8} Posts 
I was thinking that the two programs could be combined with that check. With two factors I imagine we could come up with an alternative formula.

20210324, 22:42  #1036 
"Oliver"
Sep 2017
Porta Westfalica, DE
7×71 Posts 
We can extend our procedure to all (whole nuber) bases. Since I did some research since last time and I've not been following the Aliquot subproject forever, I'd like to preface that I use \(\sigma\) as the aliquot sum function, not the sum of divisors function (i.e. in \(\sigma(n)\), the \(n\) addend is missing).
For \(x := \prod_{i=0}^m{p_i}\), where \(p_i, i \in \{0, \dots, m\}\) are all prime but not necessarily distinct and \(1 < x \in \mathbb{N}\), we have \[\sigma(x^n) = \left(\prod_{i=0}^m \frac{p_i^n1}{p_i1}\right)  x^n = \frac{\prod_{i=0}^m{\left(p_i^n1\right)}}{\prod_{i=0}^m{\left(p_i1\right)}}  x^n = \frac{\sum_{j=0}^{2^m1}{\left(1^{m  \text{popcnt}(j)} \cdot \prod_{i=0}^m{p_i^{n \cdot \left\lfloor \frac{j \bmod 2^{i+1}}{2^i} \right\rfloor}}\right)}}{\prod_{i=0}^m{\left(p_i1\right)}}  x^n,\] and that way we might only have to do modular exponentiations, but we have \(2^m\) addends of them. So for bases with a small amount of prime factors this should be feasible. The part where I get to the exponent with a fraction of something divided by the floor of something is binary thinking to expand the \(\prod_{i=0}^m{\left(p_i^n1\right)}\) part, \(\left\lfloor \frac{j \bmod 2^{i+1}}{2^i} \right\rfloor\) checks for a bit being set; I'll have to revisit that tomorrow or so and write some code. That way I'm sure I can come up with a much more stringent explanation. I'm not yet totally sure about the equation I gave above (at least I'm totally positive!), but I'm sure that we'll be getting somewhere with this. It's late for me today, and as we say "nach müde kommt blöd" ("you'll be stupid when you are too tired"), that's where we had a problem with our chancellor yesterday. 
20210325, 18:08  #1037 
"Alexander"
Nov 2008
The Alamo City
3·7·29 Posts 
I went ahead and compiled together the conjecture web page based on the attachment in an earlier post. I kept the full section titles rather than using the CSS counters in this version, though that can be adjusted. I moved the conjecture list to the beginning, but I didn't really touch the rest of the content other than shoving it to the end. JeanLuc, you can handle that content as you see fit, as my original idea didn't have that extra stuff on the page (it was basically just the conjecture list, as the rest can be found on this thread). I added some formatting and a couple of easy proofs as examples. Let me know what you think.

20210325, 20:43  #1038 
"Garambois JeanLuc"
Oct 2011
France
3^{2}×5×13 Posts 
@Happy5214 :
A very very big thank you Happy for this work ! I haven't had time to work on this page yet. I will try to review your page in detail next weekend, as I will not need to update the reservation page. But I think it's a good idea to present all of the conjectures like this. The idea of putting proofs with the color code green red also appeals to me. However, I don't know how to link to a given post number, since as you told me one day, not everyone has the same number of posts per page. Does anyone know how to do this ? @kruoli : Thank you very much for your interest in our project. But in my case, I am not able to answer your posts : I do not have the level in mathematics. So if I don't respond to your posts, it's not for lack of interest, on the contrary. Other people here are much more skilled than I at handling such formulas. Having said that, I find it absolutely wonderful that our project has given rise to so many discussions. I will try to understand your steps and maybe soon I will even be able to use your results for my own conjectures research ... 
20210326, 02:17  #1039  
"Ed Hall"
Dec 2009
Adirondack Mtns
2^{3}×3×5×31 Posts 
Quote:
Example: https://www.mersenneforum.org/showpo...postcount=1038 is your previous post. Last fiddled with by EdH on 20210326 at 02:18 

20210326, 16:32  #1040 
"Garambois JeanLuc"
Oct 2011
France
249_{16} Posts 
OK, thank you very much Edwin for these explanations.

20210326, 22:43  #1041 
"Oliver"
Sep 2017
Porta Westfalica, DE
1F1_{16} Posts 
Wait a bit, please. I greatly overcomplicated what needs to be done and there are (at least...) two errors in the last formula I posted. I'll update you when I have a running program. It is already implemented and works on some given inputs, but I need to do some more testing. It will be written in C# since I am most familiar with it and I'll share the code so that whoever wants to can port it to GMP code. It will have vast room for improvement but is already relatively fast for the inputs that are used in this project.

20210328, 12:07  #1042  
"Garambois JeanLuc"
Oct 2011
France
3^{2}·5·13 Posts 
Quote:
OK. What do you think of this page for publication on my website, as an attached file ? I hope I haven't sabotaged your page with my edits ! PS : I draw your attention to the conjectures demonstrated on the forum, conjecture (2) and conjecture (138) and to the additional information at the bottom : should I leave it ? 

20210328, 16:55  #1043  
"Alexander"
Nov 2008
The Alamo City
261_{16} Posts 
Quote:
Contentwise, my preference is that the proofs be added to the page itself rather than included using links to forum posts. That way, we can clean them up and keep them all in one place. In terms of HTML code: the <h4> tags in the last two sections need to be promoted to <h3>, since there are no intermediate <h3> levels like there are in the first section. h3 will have to be added to the CSS styling. You also left off <section> tags for the new headers. And you can't legally write a bare ampersand ("&"), as that's HTML's escape character. Use "&" instead. 

20210330, 20:28  #1044 
"Oliver"
Sep 2017
Porta Westfalica, DE
7·71 Posts 
PATF tool
Here I have my trial factoring tool for \(\sigma(n^m)\) for very large \(m\). I wrote it in C#, wanted to use .NET Core originally, but found .NET Framework easier to use on Linux with Mono. I have no experience with .NET Core on Linux.
It should be reasonably fast, but has tons of room for improvement, both algorithmically and programminglanguage wise. Usage (Windows): Code:
patf {base} {exponent} {trial factoring limit} Code:
mono patf.exe {base} {exponent} {trial factoring limit} Code:
> patf 7 1987441237556775 1000000 3^5 19^2 29^2 31 37^2 47 59 83 103 109 131 139 ... (snipped here, there would all show up normally) 908821 909151 914629 919081 920323 929881 949051 956341 958393 963901 967847 982801 991453 476 factors found. 469 factors are unique. Preparations took 2 ms, factoring of the base took 6 ms, trying factor candidates took 1,114 ms. Last fiddled with by kruoli on 20210330 at 21:03 Reason: Corrected example formula. Corrected and clarified example. 
20210331, 13:39  #1045  
"Garambois JeanLuc"
Oct 2011
France
3^{2}×5×13 Posts 
Quote:
Quote:
Please, if someone can check the proofs of conjectures (2) and (138), if it fits like this ? I still left the links to the posts. Moreover, I don't know how we usually do it : do I have to add the names of the people who made the demonstrations each time, which seems correct to me ? (conjecture 2: henryzz and warachwe ; conjecture 3: henryzz, warachwe, Happy5214 ; other conjectures : Happy5214) For conjecture 138, should I add the proof of Happy5214 from post #960 ? Quote:
However, there is no reason that the size of these titles should be different, in my opinion. I'm really sorry, but I don't understand what to do here ? I hope this is good now, because I only half understand the usefulness of sections and how they work ! Quote:
Because here too I do not understand what to do ? The simplest is perhaps that for this kind of details which I do not know, you make the modifications directly on the attached file and that you rename it "conjectures4" in order to send it back to me. Only do this if it is important to you, because personally, I am more than satisfied with the page as it is in this version 3 ... And I'm sorry to take your time like this, because I think you have a lot of other things to do ! 

Thread Tools  
Similar Threads  
Thread  Thread Starter  Forum  Replies  Last Post 
Broken aliquot sequences  fivemack  FactorDB  46  20210221 10:46 
Broken aliquot sequences  schickel  FactorDB  18  20130612 16:09 
A new theorem about aliquot sequences  garambois  Aliquot Sequences  34  20120610 21:53 
poaching aliquot sequences...  Andi47  FactorDB  21  20111229 21:11 
New article on aliquot sequences  schickel  mersennewiki  0  20081230 07:07 