20151122, 10:03  #1 
Basketry That Evening!
"Bunslow the Bold"
Jun 2011
40<A<43 89<O<88
3·29·83 Posts 
Introduction to Aliquot Analysis
In the last few weeks I've finally come to fully understand Clifford Stern's work  it took me quite a lot of effort of working out a lot of significant details that Stern's page seems to otherwise assume that you already know. And the other thread I created recently had other related questions I was asking myself and have worked out since, with hints and help from the forum members.
Having in fact worked out all the details, in retrospect it would have been nice if someone presented the missing details along with the stuff that Stern was talking about. So that's what I've done. I've made a large new theory page that starts from the very basics, defining what an aliquot sequence is etc, and working its way all the way up through the entirety of Clifford Stern's definitions and conclusions, except presented in a pedagogically oriented way (I hope). That is, all the definitions are properly motivated, as opposed to saying "here's a definition, and here's why it's important" as if some math god knew ahead of time why it's important and where to look to solve such matters. (I've also introduced a bit of new notation that helped simplify my understanding of things.) Although the page includes almost no theory beyond Clifford's material, some here may yet find it educational, and I'd appreciate it if even the experts read it and gave me feedback. (Yes, the formatting is awful with all inline equations, and yes there could easily be a hundred links in the page, but I've only got so much time each day. I'm more interested in pedagogical feedback ) Also there's a small side problem which I've been unable to resolve, and everyone's thoughts on the matter are quite welcome (especially you experts please!) Edit: Augh! The main page: https://htmlpreview.github.io/?https...analysis.html (soon to be available here when the rechencraft server pulls the latest commits) And the small side problem with which I would appreciate some external insight: https://htmlpreview.github.io/?https...imepowers.html (again soon to be here) Last fiddled with by Dubslow on 20151122 at 10:11 
20151128, 10:40  #2 
May 2004
New York City
5·7·11^{2} Posts 
Just starting to read your refreshing work.
Two paragraphs in and it looks and reads clean. 
20151128, 11:04  #3 
May 2004
New York City
4235_{10} Posts 
In section 2:
where you have Code:
prod of asubi + 1 Code:
prod of (asubi + 1) 
20151128, 11:17  #4  
Basketry That Evening!
"Bunslow the Bold"
Jun 2011
40<A<43 89<O<88
3×29×83 Posts 
Quote:
Quote:
Please. Though perhaps the thread in the aliquot forum is more appropriate? Last fiddled with by Dubslow on 20151128 at 11:38 

20151128, 12:01  #5  
May 2004
New York City
108B_{16} Posts 
Quote:
I just happened on your post here, and so started reading your aliquot paper version, and found this possible typo. I may look into more later. 

20151128, 18:53  #6  
May 2004
New York City
5×7×11^{2} Posts 
Quote:
The document is yours, of course. The next typo is in (ptothea minus 1) / (p  1) [ you wrote b for a ]. I'll stop now. Wish you success. Last fiddled with by davar55 on 20151128 at 18:57 

20151129, 05:06  #7 
Basketry That Evening!
"Bunslow the Bold"
Jun 2011
40<A<43 89<O<88
3·29·83 Posts 
You still see missing parens? Where?
The a>b typo's been fixed 
20151129, 11:43  #8  
"Alexander"
Nov 2008
The Alamo City
2·3^{3}·17 Posts 
Introduction to Aliquot Analysis
Quote:


20151129, 12:23  #9  
Basketry That Evening!
"Bunslow the Bold"
Jun 2011
40<A<43 89<O<88
3·29·83 Posts 
Quote:
http://htmlpreview.github.io/?http:/...imepowers.html "But in fact, in my numerical tests, I have been unable to find a counter example where τ(p^a)≠τ(p)+β(l) [where a=2l1], where the search extended to p<10^9, and each prime was tested up to a=99." ^ said tests were run in Python, and they're what made me look for the proof at all (which axn provided ). In fact, all code I've ever written for aliquot sequences, including the website, is written in Python. Last fiddled with by Dubslow on 20151129 at 12:24 

20151129, 13:54  #10 
May 2004
New York City
5×7×11^{2} Posts 

20151129, 22:44  #11 
Just call me Henry
"David"
Sep 2007
Liverpool (GMT/BST)
37×163 Posts 
I am slowly working through your page. It looks very good so far.
Thought I would point out another form of guide: Take for example . This means that . This process can be repeated to get which can't loose 7, 73, 37, 19, 5 or 3 as a factor. These can be fairly vulnerable to getting squared terms and then loosing the guide but they are fairly stable and are highly abundant. Here is a fairly good example: http://factordb.com/sequences.php?se...nge&fr=0&to=30 Can mutate directly to the form ? It rings a bell that there are restrictions on how 2^3*3*5*n can mutate. Forgive me if this is mentioned. I have only properly read up to section 3 and glanced at the rest. 
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