Aliquot sequences that start on the integer powers n^i - Page 46

EdH · 2020-08-24, 19:47

Here's a full list of all the terminating primes from all the sequences across all the tables on the page. I also added base 79 through power 79 since it's ready to add to the page. There are 1692 unique terminating primes found to this date.

I still haven't solved the exponent sorting, but the bases should be sorted.

EdH · 2020-08-25, 01:39

Quote:

Originally Posted by garambois

. . .
I'm also wondering why from the prime number 337, we can't find the factorizations of the exponent anymore ?
. . .

Although they are there for further, they are still missing after a point. I am trying to find out why, but not getting anywhere, yet. Oddly they stop during the 601 prime, at the line right after the 601 line of the document.

EdH · 2020-08-25, 02:04

Quote:

Originally Posted by EdH

Although they are there for further, they are still missing after a point. I am trying to find out why, but not getting anywhere, yet. Oddly they stop during the 601 prime, at the line right after the 601 line of the document.

Alright! I found the culprit. I think this document will be complete. The 601 was a subtle indicator - an easy fix after all, once I realized the real problem.

RichD · 2020-08-25, 02:32

Table n=20 will be ready to insert about the time the next update is due to be posted.

garambois · 2020-08-25, 09:39

@EdH : Many thanks to you. The table is perfect. All I have to do now is to do by hand, the last sort in ascending order of exponents, while waiting to write my own program that will give the same output.
But for the moment at the sight of this table, it does not seem to appear of new conjecture !
Unfortunately, the start of the school year is fast approaching and I would have less time to work on on the aliquot sequences !

@RichD : Thanks a lot. I don't think I'll do an update until next weekend. I will then add bases 79 and 20...

garambois · 2020-08-25, 10:50

About the search for possible Open-End sequences among those that should end trivially

The sequences we have calculated so far that end trivially are those of bases 2, 18 and those beginning with b^i with b (base) and i (exponent) of the same parity. For all these sequences, at the first iteration, we get an odd number. This is what makes them end. All the following terms remain essentially odd, with rare exceptions where we always end up with an odd number. For one of these aliquot sequence to be Open-End, one of its odd terms would have to be an aliquot antecedent of an even Open-End aliquot sequence, which we are familiar with in the main project.

Let me give you the following two facts, and after, draw the conclusions :

1) At the last update on August 21, 2020, in our project, we had calculated 1827 sequences that ended trivially.

2) Using my fundamental database, I determined the following :
- There are 4 odd numbers less than 10,000 which are the start of Open-End sequences (3025, 7225, 8015 and 8427)
- There are 80 odd numbers less than 100,000 which are the start of Open-End sequences.
- There are 810 odd numbers less than 1,000,000 which are the start of Open-End sequences.
- There are 7734 odd numbers less than 10,000,000 which are the start of Open-End sequences.
In fact, these numbers are a bit too big, because in my fundamental database, a sequence is Open-End as soon as its terms reach 50 digits.
But the order of magnitude is there: out of all the odd numbers, there is 1 in 1300 or even say 1 in 2000, which is the start of an Open-End sequence.

Taking into account these two facts, and as in a sequence, one "picks" an odd number at each iteration, I ask the question that causes me so much trouble :

Why hasn't an Open-End sequence been found yet among those that must end trivially ?

garambois · 2020-08-25, 12:30

Please, on the previous post, I spotted an error and I can no longer correct myself !

Code:

- There are 810 odd numbers less than 100,000 which are the start of Open-End sequences.
- There are 7734 odd numbers less than 1,000,000 which are the start of Open-End sequences.
In fact, these numbers are a bit too big, because in my fundamental  database, a sequence is Open-End as soon as its terms reach 50 digits.
But the order of magnitude is there: out of all the odd numbers, there  is 1 in 130 or even say 1 in 200, which is the start of an Open-End  sequence.

Should be :

Code:

- There are 810 odd numbers less than 1,000,000 which are the start of Open-End sequences.
- There are 7734 odd numbers less than 10,000,000 which are the start of Open-End sequences.
In fact, these numbers are a bit too big, because in my fundamental  database, a sequence is Open-End as soon as its terms reach 50 digits.
 But the order of magnitude is there: out of all the odd numbers, there  is 1 in 1300 or even say 1 in 2000, which is the start of an Open-End  sequence.

Many thanks !

garambois · 2020-08-25, 17:30

Thank you so much Ed !

yoyo · 2020-08-25, 19:41

I take base 12.
Do I take too much?
On base 13 are some reservations.

VBCurtis · 2020-08-25, 19:51

I've been working on base 13 for a few years, and prefer to work only on base 13. Please skip that one!

EdH · 2020-08-25, 20:30

OK! I think I have them all sorted correctly, now. Attached is a new document with bases AND powers sorted (and, the power expansion is to the end).

2020-08-25, 10:50	#501
garambois "Garambois Jean-Luc" Oct 2011 France 1010001001₂ Posts	About the search for possible Open-End sequences among those that should end trivially The sequences we have calculated so far that end trivially are those of bases 2, 18 and those beginning with b^i with b (base) and i (exponent) of the same parity. For all these sequences, at the first iteration, we get an odd number. This is what makes them end. All the following terms remain essentially odd, with rare exceptions where we always end up with an odd number. For one of these aliquot sequence to be Open-End, one of its odd terms would have to be an aliquot antecedent of an even Open-End aliquot sequence, which we are familiar with in the main project. Let me give you the following two facts, and after, draw the conclusions : 1) At the last update on August 21, 2020, in our project, we had calculated 1827 sequences that ended trivially. 2) Using my fundamental database, I determined the following : - There are 4 odd numbers less than 10,000 which are the start of Open-End sequences (3025, 7225, 8015 and 8427) - There are 80 odd numbers less than 100,000 which are the start of Open-End sequences. - There are 810 odd numbers less than 1,000,000 which are the start of Open-End sequences. - There are 7734 odd numbers less than 10,000,000 which are the start of Open-End sequences. In fact, these numbers are a bit too big, because in my fundamental database, a sequence is Open-End as soon as its terms reach 50 digits. But the order of magnitude is there: out of all the odd numbers, there is 1 in 1300 or even say 1 in 2000, which is the start of an Open-End sequence. Taking into account these two facts, and as in a sequence, one "picks" an odd number at each iteration, I ask the question that causes me so much trouble : Why hasn't an Open-End sequence been found yet among those that must end trivially ? Last fiddled with by EdH on 2020-08-25 at 13:14 Reason: See next post.

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2020-08-25, 02:32	#499
RichD Sep 2008 Kansas D3D₁₆ Posts	Table n=20 will be ready to insert about the time the next update is due to be posted.

2020-08-25, 09:39	#500
garambois "Garambois Jean-Luc" Oct 2011 France 1010001001₂ Posts	@EdH : Many thanks to you. The table is perfect. All I have to do now is to do by hand, the last sort in ascending order of exponents, while waiting to write my own program that will give the same output. But for the moment at the sight of this table, it does not seem to appear of new conjecture ! Unfortunately, the start of the school year is fast approaching and I would have less time to work on on the aliquot sequences ! @RichD : Thanks a lot. I don't think I'll do an update until next weekend. I will then add bases 79 and 20...

2020-08-25, 17:30	#503
garambois "Garambois Jean-Luc" Oct 2011 France 11×59 Posts	Thank you so much Ed !

2020-08-25, 19:41	#504
yoyo Oct 2006 Berlin, Germany 623₁₀ Posts	I take base 12. Do I take too much? On base 13 are some reservations.

2020-08-25, 19:51	#505
VBCurtis "Curtis" Feb 2005 Riverside, CA 11·443 Posts	I've been working on base 13 for a few years, and prefer to work only on base 13. Please skip that one!