Aliquot sequences that start on the integer powers n^i

[RichD]

Next is table 58.

[EdH]

Originally Posted by RichD

Next is table 58.

Base 58 has the following merge so far:

 
58^3:i63 merges with 65436:i2

[Happy5214]

Originally Posted by henryzz

I think the trick would be to prove that:
2^(2^3 * 3^2 * 5 * 7 * k + 1) - 1 == 1 mod d
3^(2^3 * 3^2 * 5 * 7 * k + 1) - 1) / 2 == 1 mod d
6^(2^3 * 3^2 * 5 * 7 * k) == 1 mod d

If this is true then s(6^(2^3 * 3^2 * 5 * 7 * k)) = (2^(2^3 * 3^2 * 5 * 7 * k + 1) - 1)*3^(2^3 * 3^2 * 5 * 7 * k + 1) - 1) / 2 - 6^(2^3 * 3^2 * 5 * 7 * k) = 1*1-1 mod d

To complete and streamline warachwe's proof, it's all provable by induction using . Note that the exponent constant resolves to 2520. 6^(2^3 * 3^2 * 5 * 7 * k) == 1 mod d relatively trivially (6^2520 = 1 mod d, and 1^k = 1 mod d for all integers k). To prove base 2, I translate it to the equivalent problem of solving 2^(2^3 * 3^2 * 5 * 7 * k + 1) = 2 mod d. In this case, 2^2521 (base case) = 2 mod d, and when 2^(2520*k+1) = 2 mod d, 2^(2520*(k+1)+1) = 2 mod d (since 2^2520 = 1 mod d and 2^(2520*(k+1)+1) = 2^(2520*k+1) * 2^2520). Similarly, 3^(2520*k+1) = 3 mod d for all integers k, a statement equivalent to the above formula in , so the three formulas are proved.

[Happy5214]

@Jean-Luc: Any word on the redesigned web pages (either the sequence data or the conjectures)? As an aside, do you have access to any server-side scripting languages on your website? I still think a static page generated by an offline script is the way to go given the amount of data, especially given the way the sequence data is collected, but I am curious. In fact, a script to generate the conjecture page might be better than a handwritten HTML page, since it could automatically number the conjectures. In my mock-up, that's done by a CSS counter, which while convenient unfortunately takes most of the header content (like the number and the word "conjecture") out of the DOM and is bad for semantics and screenreaders.

I do have an improvement to the generator script for the sequence page. Does the script cache sequences that have been terminated? If not, it could save time by not requesting those from FactorDB every time and just using the cached data, which will not change again once terminated.

[garambois]

Originally Posted by Happy5214

To complete and streamline warachwe's proof, it's all provable by induction using . Note that the exponent constant resolves to 2520. 6^(2^3 * 3^2 * 5 * 7 * k) == 1 mod d relatively trivially (6^2520 = 1 mod d, and 1^k = 1 mod d for all integers k). To prove base 2, I translate it to the equivalent problem of solving 2^(2^3 * 3^2 * 5 * 7 * k + 1) = 2 mod d. In this case, 2^2521 (base case) = 2 mod d, and when 2^(2520*k+1) = 2 mod d, 2^(2520*(k+1)+1) = 2 mod d (since 2^2520 = 1 mod d and 2^(2520*(k+1)+1) = 2^(2520*k+1) * 2^2520). Similarly, 3^(2520*k+1) = 3 mod d for all integers k, a statement equivalent to the above formula in , so the three formulas are proved.

OK, so I understand that conjecture (138) is proven.
Thanks to you for all this work !

[garambois]

Originally Posted by Happy5214

@Jean-Luc: Any word on the redesigned web pages (either the sequence data or the conjectures)? As an aside, do you have access to any server-side scripting languages on your website? I still think a static page generated by an offline script is the way to go given the amount of data, especially given the way the sequence data is collected, but I am curious. In fact, a script to generate the conjecture page might be better than a handwritten HTML page, since it could automatically number the conjectures. In my mock-up, that's done by a CSS counter, which while convenient unfortunately takes most of the header content (like the number and the word "conjecture") out of the DOM and is bad for semantics and screenreaders.

I saw in your page, the automatic counter for the conjectures.
And I will make this conjectures summary page on my website according to your example.
However, I am very short on time.
Right now I'm working on conjectures and programs.
I also have to update the page regularly, which takes a long time.
And since tomorrow I'm going back to work, I don't know when I'll do the conjectures page on my website : at worst, during my next vacation in mid April.
But all these tasks are noted and pending ...

Originally Posted by Happy5214

I do have an improvement to the generator script for the sequence page. Does the script cache sequences that have been terminated? If not, it could save time by not requesting those from FactorDB every time and just using the cached data, which will not change again once terminated.

I am totally unable to answer this question.
I'll let Karsten answer, if he sees this message and has a response.
But I think right now he's working on it and maybe he's thinking about it all.
In the meantime, I am working with the old scripts and the old page.

[garambois]

Page updated.
Thanks to all for your help !

Added bases : 38 and 46.
New cycles : 39^9 and 439^46.

73 bases in total.

I also corrected a reservation error for base 15015.
I had reserved it for yoyo by mistake.
I corrected and now it is reserved for me.

[kar_bon]

Originally Posted by Happy5214

Does the script cache sequences that have been terminated? If not, it could save time by not requesting those from FactorDB every time and just using the cached data, which will not change again once terminated.

The current script re-read only open sequences from FactorDB. Terminated seqs are collected with data of the last index separately.

[yoyo]

I'll take bases 38, 43, 46, 47.

yoyo

[garambois]

Originally Posted by yoyo

I'll take bases 38, 43, 46, 47.

yoyo

OK thanks a lot.
This will be done in the next update.

[RichD]

Might have a merge with 58^7.