mersenneforum.org Aliquot sequences that start on the integer powers n^i
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2021-03-24, 21:24   #1035
henryzz
Just call me Henry

"David"
Sep 2007
Cambridge (GMT/BST)

133568 Posts

Quote:
 Originally Posted by Happy5214 Given the name, audience, use case, and lack of actual values to use it on, I figured it was unnecessary. I guess not. I'll add it (which is, what, 4 lines?) when I get back to my main computer tomorrow. Until then, do not pass it a composite base.
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.

2021-03-24, 22:42   #1036
kruoli

"Oliver"
Sep 2017
Porta Westfalica, DE

7×71 Posts

Quote:
 Originally Posted by Happy5214 (it only works with primes right now)
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
$\sigma(x^n) = \left(\prod_{i=0}^m \frac{p_i^n-1}{p_i-1}\right) - x^n = \frac{\prod_{i=0}^m{\left(p_i^n-1\right)}}{\prod_{i=0}^m{\left(p_i-1\right)}} - x^n = \frac{\sum_{j=0}^{2^m-1}{\left(-1^{m - \text{popcnt}(j)} \cdot \prod_{i=0}^m{p_i^{n \cdot \left\lfloor \frac{j \bmod 2^{i+1}}{2^i} \right\rfloor}}\right)}}{\prod_{i=0}^m{\left(p_i-1\right)}} - x^n,$
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 $$\prod_{i=0}^m{\left(p_i^n-1\right)}$$ 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.

2021-03-25, 18:08   #1037
Happy5214

"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. Jean-Luc, 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.
Attached Files
 conjectures1.tar.gz (11.1 KB, 16 views)

2021-03-26, 02:17   #1039
EdH

"Ed Hall"
Dec 2009

23×3×5×31 Posts

Quote:
 Originally Posted by garambois . . . 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 ? . . .
If you want the hyperlink for an individual post, you can hover over the post number on your view and use the right-click menu to copy the link address, or you can click the post number and then copy from the address bar. Posts per page doesn't affect the post number.

Example: https://www.mersenneforum.org/showpo...postcount=1038 is your previous post.

Last fiddled with by EdH on 2021-03-26 at 02:18

 2021-03-26, 16:32 #1040 garambois     "Garambois Jean-Luc" Oct 2011 France 24916 Posts OK, thank you very much Edwin for these explanations.
2021-03-26, 22:43   #1041
kruoli

"Oliver"
Sep 2017
Porta Westfalica, DE

1F116 Posts

Quote:
 Originally Posted by garambois I will try to understand your steps and maybe soon I will even be able to use your results for my own conjectures research ...
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.

2021-03-28, 12:07   #1042
garambois

"Garambois Jean-Luc"
Oct 2011
France

32·5·13 Posts

Quote:
 Originally Posted by Happy5214 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. Jean-Luc, 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.

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 ?
Attached Files
 conjectures2.html.tar.gz (11.6 KB, 12 views)

2021-03-28, 16:55   #1043
Happy5214

"Alexander"
Nov 2008
The Alamo City

26116 Posts

Quote:
 Originally Posted by garambois 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 ?
To answer your last question first, I'd remove that stuff unless there's something you want to do with that info. It's all available in this thread, and it doesn't really benefit the conjecture list.

Content-wise, 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 "&amp;" instead.

2021-03-30, 20:28   #1044
kruoli

"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 programming-language wise.

Usage (Windows):
Code:
patf {base} {exponent} {trial factoring limit}
Usage (Linux with Mono):
Code:
mono patf.exe {base} {exponent} {trial factoring limit}
Example (get factors of $$\sigma(7^{1987441237556775})$$ up to 1,000,000):
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.
The attachment contains both the source and an executable for usage on both Windows and Linux 64 bit, but might also run on 32 bit OSes (untested).
Attached Files
 patf.7z (13.3 KB, 14 views)

Last fiddled with by kruoli on 2021-03-30 at 21:03 Reason: Corrected example formula. Corrected and clarified example.

2021-03-31, 13:39   #1045
garambois

"Garambois Jean-Luc"
Oct 2011
France

32×5×13 Posts

Quote:
 Originally Posted by Happy5214 To answer your last question first, I'd remove that stuff unless there's something you want to do with that info. It's all available in this thread, and it doesn't really benefit the conjecture list.
OK, done.

Quote:
 Originally Posted by Happy5214 Content-wise, 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.
OK, done.
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:
 Originally Posted by Happy5214 In terms of HTML code: the

tags in the last two sections need to be promoted to

, since there are no intermediate

levels like there are in the first section.

I tried to do this, but what bothers me is that in this case the write size of the titles "Conjectures (134)" to "Conjectures (139)" is then larger than the write size of the titles "Conjectures (1)" to "Conjectures (133)".
However, there is no reason that the size of these titles should be different, in my opinion.

Quote:
 Originally Posted by Happy5214 h3 will have to be added to the CSS styling.
I'm really sorry, but I don't understand what to do here ?

Quote:
 Originally Posted by Happy5214 You also left off
I hope this is good now, because I only half understand the usefulness of sections and how they work !

Quote:
 Originally Posted by Happy5214 And you can't legally write a bare ampersand ("&"), as that's HTML's escape character. Use "&" instead.
Are you talking about the ampersands that are in the links that lead to other sites ?
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 !

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