20200710, 15:39  #353 
Oct 2011
337 Posts 
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 ! 
20200710, 17:02  #354  
"Ed Hall"
Dec 2009
Adirondack Mtns
17×197 Posts 
Quote:
I'll upload the other abundance/deficiency files later today. 

20200710, 17:42  #355 
"Ed Hall"
Dec 2009
Adirondack Mtns
17·197 Posts 
Here are bases 10 through 14:

20200710, 17:45  #356 
"Ed Hall"
Dec 2009
Adirondack Mtns
17×197 Posts 
Here are bases 15 through 28:

20200711, 09:29  #357 
Oct 2011
337 Posts 
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 ! 
20200711, 11:39  #358 
Oct 2011
337 Posts 
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^(2690000269) 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... 
20200711, 15:40  #359 
"Alexander"
Nov 2008
The Alamo City
178_{16} Posts 
Not conjecture, theorem. If p = ab (a, b > 1), then 2^a1 and 2^b1 both divide 2^p1. Ergo, any number that divides 2^n1 will also divide 2^(ni)1, for any i ≥ 1. That's why exponents for Mersenne primes must themselves also be prime.
Last fiddled with by Happy5214 on 20200711 at 15:45 
20200711, 16:21  #360 
Oct 2011
521_{8} Posts 
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 ! 
20200711, 17:51  #361 
Oct 2011
337 Posts 
OK, page updated.
Base 30 added. A lot of thanks to all. The next update will not occur until early August. 
20200711, 18:16  #362 
"Ed Hall"
Dec 2009
Adirondack Mtns
17·197 Posts 
Have a great trip! I'll see if I can form some intriguing questions while you're away.

20200711, 19:39  #363  
Oct 2011
337 Posts 
Quote:
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/c2020f3info 

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