Aliquot sequences that start on the integer powers n^i

[EdH]

Originally Posted by garambois

. . .
@ Everyone :
For the moment, I'm not making any more conjectures like the ones in post #447. I'm trying to spot other different things, especially concerning the occurrences of prime numbers that end sequences. And also, for n=2^i, for n=18^i, for n=m^i, with m and i in the same parity, I know that s(n) is odd and therefore the sequence that starts with n most likely ends. But, I don't understand why all these sequences without exception end. I'm looking forward to the first Open-End sequence for n=2^i, for n=18^i, for n=m^i, with m and i in the same parity ! But we'll have to wait until the amount of data we have increases. But with the computing power available to us at the moment, I estimate that in one week, we'll make more progress than in several years at the rate of a few months ago !
That said, I have no idea what "enough data" means. Is it going to take several months, a year, or more ?
. . .

I have extended base 2 up through i=549. I can work a few more there (not sure if I'll make it to 559), but if yoyo wants to take over with base 2, that would be fine.

I'm working on some scripts/programs that will list all the terminating primes for a base, such as I supplied for base 79. I will be completing these for all the bases in the table. I have the scripts written such that when it runs, if the local .elf does not show a termination, the script downloads a fresh .elf. That ensures that any new terminations should be caught. Once I have a set of all the terminations for all the tables, we should be able to cross-match as well.

Let me know where you would prefer I work and anything you might like me to try to harvest with my scripts/programs.

[garambois]

Thank you for extending the calculations for Base 2. I'll make the change at the next update. It's a fundamental base.

It's a great idea to create all the lists of prime numbers at the end of the sequences of all the bases, like for base 79 ! I'm working on that too.
The main problem is to display everything as legibly as possible, so that we can see things. I still don't know which elements are important !
Maybe you need to show the prime exponent factorizations for each base or something like that ? It's up to us to be creative...
Please, if it's possible, for example for base 79, display it more like this, so that we can see more information at only a glance :

7 (1) 
11 (1) 
13 (1) 
19 (1) 
23 (1)
...
41 (3)
...

should be :

7 (i1=p1^a1*p2^a2*...) 
11 (i1=p1^a1*p2^a2*...)
13 (i1=p1^a1*p2^a2*...) 
19 (i1=p1^a1*p2^a2*...) 
23 (i1=p1^a1*p2^a2*...)
...
41 (i1=p1^a1*p2^a2*...)
41 (i2=p1^a1*p2^a2*...)
41 (i3=p1^a1*p2^a2*...)
...

where i1, i2, i3 are, for example, the three exponents of the sequences ending with the prime number 41 and for each of these three exponents its factorization in prime numbers appears.

[EdH]

Originally Posted by garambois

Thank you for extending the calculations for Base 2. I'll make the change at the next update. It's a fundamental base.

It's a great idea to create all the lists of prime numbers at the end of the sequences of all the bases, like for base 79 ! I'm working on that too.
The main problem is to display everything as legibly as possible, so that we can see things. I still don't know which elements are important !
Maybe you need to show the prime exponent factorizations for each base or something like that ? It's up to us to be creative...
Please, if it's possible, for example for base 79, display it more like this, so that we can see more information at only a glance :

7 (1) 
11 (1) 
13 (1) 
19 (1) 
23 (1)
...
41 (3)
...

should be :

7 (i1=p1^a1*p2^a2*...) 
11 (i1=p1^a1*p2^a2*...)
13 (i1=p1^a1*p2^a2*...) 
19 (i1=p1^a1*p2^a2*...) 
23 (i1=p1^a1*p2^a2*...)
...
41 (i1=p1^a1*p2^a2*...)
41 (i2=p1^a1*p2^a2*...)
41 (i3=p1^a1*p2^a2*...)
...

where i1, i2, i3 are, for example, the three exponents of the sequences ending with the prime number 41 and for each of these three exponents its factorization in prime numbers appears.

I'm not sure I understand this fully at first glance, but I'll work on it.

For now, attached is a listing for all primes <1000 across all the tables, showing which sequences they terminate, if any. For example, here's a sample:

Prime 2:
============
2^1

Prime 3:
============
2^164
2^2
2^305
2^317
2^4
2^55
3^1
3^247
3^2
3^5
5^38
6^152
7^4
7^77
11^15
11^2
12^1
12^2
13^15
14^76
14^80
15^1
19^15
21^21
21^55
23^3
30^1

Prime 5:
============
5^1

Prime 7:
============
2^10
2^12
2^141
. . .
Prime 41:
============
2^112
2^117
2^23
2^281
2^373
2^405
2^411
2^47
2^6
2^8
6^13
6^5
7^11
11^14
11^57
12^32
12^56
15^2
15^6
17^65
18^39
24^18
79^17
79^2
79^4
210^4
385^11
439^2
770^13
2310^8
. . .
Prime 149:
============
3^55

Prime 151:
============
18^79
210^40

Prime 157:
============

Prime 163:
============
3^47

Prime 167:
============

Prime 173:
============
7^31
. . .

Sorry, they are only sorted by base, not power.

[EdH]

@Jean-Luc:

Would this be more toward what you need:

Listing of all terminating primes <1000 across all tables

Prime 2:
============
2^(1)

Prime 3:
============
2^(2^2*41)
2^(2)
2^(5*61)
2^(317)
2^(2^2)
2^(5*11)
3^(1)
3^(13*19)
3^(2)
3^(5)
5^(2*19)
6^(2^3*19)
7^(2^2)
7^(7*11)
11^(3*5)
11^(2)
12^(1)
12^(2)
13^(3*5)
14^(2^2*19)
14^(2^4*5)
15^(1)
19^(3*5)
21^(3*7)
21^(5*11)
23^(3)
30^(1)

Prime 5:
============
5^(1)

Prime 7:
============
2^(2*5)
2^(2^2*3)
2^(3*47)
2^(2*139)
. . .

I've attached a new copy of the last file of primes <1000 across all tables.

[yoyo]

Originally Posted by EdH

I have extended base 2 up through i=549. I can work a few more there (not sure if I'll make it to 559), but if yoyo wants to take over with base 2, that would be fine.

I would take base 2 from $i=540 to $i <= 559 until the remaining composite is > C139.

yoyo

[garambois]

@yoyo : Many thanks !



@EdH : Is it possible to have :

Prime 3: ============
2^164 = 2^(2^2*41) 
2^2 = 2^(2)
...

But sorted !
No problem if it's too hard to do, I'll manage otherwise!

[EdH]

Originally Posted by garambois

@yoyo : Many thanks !



@EdH : Is it possible to have :

Prime 3: ============
2^164 = 2^(2^2*41) 
2^2 = 2^(2)
...

But sorted !
No problem if it's too hard to do, I'll manage otherwise!

I was able to sort the bases, but the powers proved more difficult because the normal sort works by character, which means 164 is before 2, because 1 is before 2. I'll work on the formatting, at least.

[RichD]

Prime 3: ============
2 ^ 164 = 2^(2^2*41) 
2 ^ 2 = 2^(2)
...

Maybe add spaces and numeric sort the third field. Not pretty.

[EdH]

Here is a listing with the new formatting. I still haven't achieved the secondary (powers) sorting. These listings currently are only based on all primes below 1000. They include the mention of primes which have not been found at all as terminations. I'm running a current process to update and list only primes that are found as terminations across all tables. It's taking time to check for currency of all the sequences to see if any have terminated since the last check. Once I have the full list of all primes that terminate sequences, I will run that list for a total document of all terminating primes across all the tables on the page. That may still be a couple days.

[yoyo]

I take also base 10 and 11 until a composite is > C139.

[garambois]

Okay, thank you very much Ed.
This is exactly the document we need !
I'm also wondering why from the prime number 337, we can't find the factorizations of the exponent anymore ?

For my part, I'm currently trying to understand why we don't have an Open-End sequence for bases 2, 18 and the other bases for the exponents that have the same parity as them...