Aliquot sequences that start on the integer powers n^i

[garambois]

Originally Posted by Happy5214

I suggest you compile all of these conjectures, including the proved and disproved ones, onto a page (or pages) on your website for easy access. In particular, the link you provided in that post is irrelevant to me because I have a different number of posts per page, so I still have to search through the page list to find it, so having a single, stable link would be beneficial.

I tried to create an html page as you request.
I think it's a good idea to put these conjectures on my own website.
See attached file and give me your opinion if you wish.
You can even edit the page if you want and send it to me.

This is a copy of the forum pages presenting the conjectures.
Is it allowed to post a copy of a forum page on your own site in this way or do you have to ask someone for permission ?

On the other hand, it will be very difficult to note the conjectures already demonstrated !
To my knowledge, only one conjecture is demonstrated, it is the Conjecture (2) (see post #481).
Some of my conjectures published in August 2020 seem already known : some of those with a star, but how do I know exactly which ?
I say this by virtue of my first paragraph of post #456 which was never denied by warachwe, I quote my response to warachwe :

Originally Posted by garambois

Okay, thank you so much for your help!
I suspected this kind of result could be demonstrated.
This implies that a large part of my conjectures (almost all those with a star) are in fact proven.

[Happy5214]

Originally Posted by garambois

I tried to create an html page as you request.
I think it's a good idea to put these conjectures on my own website.
See attached file and give me your opinion if you wish.
You can even edit the page if you want and send it to me.

This is a copy of the forum pages presenting the conjectures.
Is it allowed to post a copy of a forum page on your own site in this way or do you have to ask someone for permission ?

I'm going to bed now (I have a wacky sleep schedule), so I'll look at the page when I get up. My understanding is that the conjecture forum posts in question were posted by you, so I see no problem doing whatever you want with that content.

[RichD]

Base 38 can be added at the next update. At least one merge and a cycle can be found in this table.

[EdH]

Originally Posted by RichD

Base 38 can be added at the next update. At least one merge and a cycle can be found in this table.

I didn't look for the cycle, but I found the following merges:

38^7:i166 merges with 2351700:i5
38^15:i1298 merges with 660:i21

and, the following matched parity sequences with index 2 > index 1:

38^12
38^24
38^36
38^48
38^60
38^72

[RichD]

Originally Posted by RichD

Base 38 can be added at the next update.

I took it to i=78.

[garambois]

Originally Posted by Happy5214

I'm going to bed now (I have a wacky sleep schedule), so I'll look at the page when I get up. My understanding is that the conjecture forum posts in question were posted by you, so I see no problem doing whatever you want with that content.

Have a good night !
OK for the posting rights on my website for forum excerpts written by myself, you must be right, that makes sense.
We will see if there will be a reasoned objection in the next few hours...

[garambois]

Originally Posted by RichD

Base 38 can be added at the next update. At least one merge and a cycle can be found in this table.
I took it to i=78.

OK thanks a lot.
I think I'll put base 38 up to exponent 100, as was done by yoyo for all of its reserved bases.
I have done this on the site page for all the bases less than 100 that I have added lately.

Originally Posted by EdH

I didn't look for the cycle, but I found the following merges:

38^7:i166 merges with 2351700:i5
38^15:i1298 merges with 660:i21

and, the following matched parity sequences with index 2 > index 1:

38^12
38^24
38^36
38^48
38^60
38^72

Thanks for the mergers.
Thank you for the matched parity sequences with index 2 > index 1.
I am almost certain that we will be able to add 38^84 and 38^96 !
We could also formulate a conjecture similar to Conjecture (137) for the base 38 with the exponents multiple of 12...

[EdH]

Originally Posted by garambois

. . .

Thanks for the mergers.
Thank you for the matched parity sequences with index 2 > index 1.
I am almost certain that we will be able to add 38^84 and 38^96 !
We could also formulate a conjecture similar to Conjecture (137) for the base 38 with the exponents multiple of 12...

Yes, you can:

38^12:
38^24:
38^36:
38^48:
38^60:
38^72:
38^84:
38^96:

These two last ones must have been recent. I had done base 38 through 100 originally, but my sequence downloads, done this morning, only had index 1 for ^84 and ^96. An update of my sequences now has index 2 for both.

[garambois]

I suspected it !
For 38^96, we even have a growth up to index 3.

[Happy5214]

Originally Posted by garambois

Thanks for the mergers.
Thank you for the matched parity sequences with index 2 > index 1.
I am almost certain that we will be able to add 38^84 and 38^96 !
We could also formulate a conjecture similar to Conjecture (137) for the base 38 with the exponents multiple of 12...

All of those powers seem have the factors 3*5*7*13 in the first term. While not abundant on its own (3*5*7*13 = 1365, while its aliquot sum is 1323), it is very nearly so. I wasted nearly 3 hours trying to find a formula to derive the sequence of first terms from the previous powers to write an inductive modular proof that all base 38 powers that are multiples of 12 have those factors, with no success. The rule does hold to 38^300, though, and thus they are likely to be abundant.

[Happy5214]

Originally Posted by garambois

I tried to create an html page as you request.
I think it's a good idea to put these conjectures on my own website.
See attached file and give me your opinion if you wish.
You can even edit the page if you want and send it to me.

This is a copy of the forum pages presenting the conjectures.
Is it allowed to post a copy of a forum page on your own site in this way or do you have to ask someone for permission ?

On the other hand, it will be very difficult to note the conjectures already demonstrated !
To my knowledge, only one conjecture is demonstrated, it is the Conjecture (2) (see post #481).
Some of my conjectures published in August 2020 seem already known : some of those with a star, but how do I know exactly which ?
I say this by virtue of my first paragraph of post #456 which was never denied by warachwe, I quote my response to warachwe :

You know, you don't have to copy the forum posts exactly. You might want to reorganize it. Maybe put the conjectures in one section and the remarks and other stuff in a separate section.