Aliquot sequences that start on the integer powers n^i

[garambois]

Yes, I'm still interested in the abundance/deficiency charts. But until base 28, that will be more than enough, no need to go any further.
Thank you very much !

Please, can you check with some base 2 exponents if you get the same thing as me, see in the attached file. That would reassure me for the future work !

[EdH]

Originally Posted by garambois

Yes, I'm still interested in the abundance/deficiency charts. But until base 28, that will be more than enough, no need to go any further.
Thank you very much !

Please, can you check with some base 2 exponents if you get the same thing as me, see in the attached file. That would reassure me for the future work !

I added the prefix you used to my output lines and ran the entire base 2 as you did. My result file was identical to yours according to diff. I would say, both our programs have been validated, or at least, they both have the same error(s).

I'll upload the other abundance/deficiency files later today.

[EdH]

Originally Posted by garambois

Yes, I'm still interested in the abundance/deficiency charts. But until base 28, that will be more than enough, no need to go any further.
Thank you very much !
. . .

Here are bases 10 through 14:

[EdH]

Originally Posted by garambois

Yes, I'm still interested in the abundance/deficiency charts. But until base 28, that will be more than enough, no need to go any further.
Thank you very much !
. . .

Here are bases 15 through 28:

[garambois]

Many thanks for all Ed !

I downloaded all the files.
And I'm very happy to learn that our programs are giving the same results !

But what surprises me is that the data analysis I'm doing for a single database takes a lot of time. I must have underestimated the amount of work the computer would need to do all the tests I planned to do.
Base 2 analysis : 8 hours.
Base 3 analysis : I started 12 hours ago and it's not finished.
I will let you know the first results before I leave, as I'm not sure I'll be able to obtain everything before my trip at this pace !

[garambois]

I think I've come up with a new conjecture ! But I would be very surprised if the experts working on the factorization of Mersenne numbers did not know this conjecture !
Thank you for keeping me informed !


Unless I'm mistaken, I'm almost certain that the prime number 68625988504811774259364670661552948915363901845035416371912463477873783063 factors all numbers of the form 2^(269*i)-1 if i is an integer. I tried on factordb to factorize 2^2690000 and 2^(2690000-269) and it worked !
It's up to you to try again.

In the same way, I think I can affirm that the prime number 160619474372352289412737508720216839225805656328990879953332340439 factorizes all numbers of the form 2^(241*i) with i integer.

And I have several more like this, see the attached file.

The attached file starts like this :

base 2 prime 10567201 exponent 75
base 2 prime 10567201 exponent 150
base 2 prime 10567201 exponent 225
base 2 prime 10567201 exponent 300
base 2 prime 10567201 exponent 375
base 2 prime 10567201 exponent 450
base 2 prime 10567201 exponent 525

This means that the prime 10567201 is a prime factor that appears in the aliquot sequences 2^75, 2^150, 2^225... and more generally 2^(75*i) with integer i. And I don't know why, but this prime number always appears in the decomposition of the number at index 1 of the sequence. So it factors the numbers 2^(75*i)-1.

I didn't check with all the prime numbers in the file, but for the ones I did, it worked like this...

So the same should happen with all the prime numbers in the file...

12112549 should factor every 2^(164*i)-1.
13264529 should factor every 2^(47*i)-1.
...
...
...
68625988504811774259364670661552948915363901845035416371912463477873783063 should factor every 2^(269*i)-1.


I'll try to do the same work with the other bases...

[Happy5214]

Not conjecture, theorem. If p = ab (a, b > 1), then 2^a-1 and 2^b-1 both divide 2^p-1. Ergo, any number that divides 2^n-1 will also divide 2^(ni)-1, for any i ≥ 1. That's why exponents for Mersenne primes must themselves also be prime.

[garambois]

Thank you very much Happy5214.
I suspected it was already known !

I continue my analysis to try to find something else on the other bases and also on a number of iterations greater than 1 to go further in the sequences ...
But I'm stuck, the data analysis by my programs for base 3 is still not finished after 20 hours of operation !

[garambois]

OK, page updated.
Base 30 added.

A lot of thanks to all.

The next update will not occur until early August.

[EdH]

Have a great trip! I'll see if I can form some intriguing questions while you're away.

[garambois]

Originally Posted by EdH

Have a great trip! I'll see if I can form some intriguing questions while you're away.

Thank you very much !
Good luck in your quest.


And don't forget to look at the Neowise Comet, it can be seen with the naked eye (but it's very low on the horizon !) : https://theskylive.com/c2020f3-info