Aliquot sequences that start on the integer powers n^i

[garambois]

I really don't have a single free thread left.
By tomorrow or Tuesday, I think I can pull out all the data I need to make the new conjectures more solid.
But that would help me if someone could still make it so that for bases 6, 12 and 30, as many sequences with even exponents as possible reach at least index 2, with the number in index 2 not necessarily factored.
For example, in the table, we see that the sequence 6^184 only arrives at index 1. It would have to indicate at least index 2. This amounts to factoring the C115.
But for the sequence 6^188, this amounts to factoring a C146, which of course poses more problems !
Be careful, no one has to drop another job to do this.
I now have a lot of data. It's just in case someone has idle threads ...
These few calculations are unlikely to change any of my conjecture. But we never know !
What I'm looking for : it's an even exponent sequence for these three bases for which the term in index 2 is greater than the term in index 1.
I haven't found any yet !

In fact, I would need the same work for all sequences with even exponents of even bases and for all sequences with odd exponents of odd bases (especially base 15).

[EdH]

Originally Posted by garambois

I really don't have a single free thread left.
By tomorrow or Tuesday, I think I can pull out all the data I need to make the new conjectures more solid.
But that would help me if someone could still make it so that for bases 6, 12 and 30, as many sequences with even exponents as possible reach at least index 2, with the number in index 2 not necessarily factored.
For example, in the table, we see that the sequence 6^184 only arrives at index 1. It would have to indicate at least index 2. This amounts to factoring the C115.
But for the sequence 6^188, this amounts to factoring a C146, which of course poses more problems !
Be careful, no one has to drop another job to do this.
I now have a lot of data. It's just in case someone has idle threads ...
These few calculations are unlikely to change any of my conjecture. But we never know !
What I'm looking for : it's an even exponent sequence for these three bases for which the term in index 2 is greater than the term in index 1.
I haven't found any yet !

In fact, I would need the same work for all sequences with even exponents of even bases and for all sequences with odd exponents of odd bases (especially base 15).

I'm working on the C115.

[EdH]

Originally Posted by EdH

I'm working on the C115.

6^184 is at index 2. . .

[garambois]

Originally Posted by EdH

6^184 is at index 2. . .

One more ! Many thanks !

[EdH]

Originally Posted by garambois

. . .
In fact, I would need the same work for all sequences with even exponents of even bases and for all sequences with odd exponents of odd bases (especially base 15).

Let me make sure I have this right:

You'd like a list showing any matched parity sequences with index 2 > index 1?

As an example, 2^12:

0 .   4096 = 2^12 
1 .   4095 = 3^2 * 5 * 7 * 13 
2 .   4641 = 3 * 7 * 13 * 17 
3 .   3423 = 3 * 7 * 163 
4 .   1825 = 5^2 * 73 
5 .   469 = 7 * 67 
6 .   75 = 3 * 5^2 
7 .   49 = 7^2 
8 .   8 = 2^3 
9 .   7 = 7

I'm working on a program to find all such sequences within the tables. Am I on the right track?

[EdH]

Originally Posted by garambois

I really don't have a single free thread left.
By tomorrow or Tuesday, I think I can pull out all the data I need to make the new conjectures more solid.
But that would help me if someone could still make it so that for bases 6, 12 and 30, as many sequences with even exponents as possible reach at least index 2, with the number in index 2 not necessarily factored.
For example, in the table, we see that the sequence 6^184 only arrives at index 1. It would have to indicate at least index 2. This amounts to factoring the C115.
But for the sequence 6^188, this amounts to factoring a C146, which of course poses more problems !
Be careful, no one has to drop another job to do this.
I now have a lot of data. It's just in case someone has idle threads ...
These few calculations are unlikely to change any of my conjecture. But we never know !
What I'm looking for : it's an even exponent sequence for these three bases for which the term in index 2 is greater than the term in index 1.
I haven't found any yet !

In fact, I would need the same work for all sequences with even exponents of even bases and for all sequences with odd exponents of odd bases (especially base 15).

Base 6, 12 and 30 so far, don't appear to have any sequences with index 2 > index 1, but base 15 has 15^35 and many others exist throughout the bases. Here's a list of all I found in the current tables:

Base 2:
2^12 
2^24 
2^36 
2^40 
2^48 
2^60 
2^72 
2^80 
2^84 
2^90 
2^96 
2^108 
2^120 
2^132 
2^140 
2^144 
2^156 
2^160 
2^168 
2^180 
2^192 
2^200 
2^204 
2^210 
2^216 
2^220 
2^228 
2^240 
2^252 
2^264 
2^270 
2^276 
2^280 
2^288 
2^300 
2^312 
2^320 
2^324 
2^330 
2^336 
2^348 
2^360 
2^372 
2^384 
2^396 
2^400 
2^408 
2^420 
2^432 
2^440 
2^444 
2^450 
2^456 
2^468 
2^480 
2^492 
2^504 
2^516 
2^520 
2^528 
2^540 
2^552 
Base 10:
10^36 
10^60 
10^72 
10^84 
10^90 
10^96 
10^108 
10^120 
10^132 
10^144 
Base 14:
14^36 
14^60 
14^72 
14^108 
14^120 
Base 15:
15^35 
Base 18:
18^120 
Base 20:
20^36 
20^42 
20^60 
20^66 
20^72 
20^90 
20^96 
Base 22:
22^6 
22^18 
22^60 
22^66 
22^100 
Base 26:
26^12 
26^18 
26^30 
26^36 
26^42 
26^54 
26^60 
26^72 
26^90 
26^96 
Base 28:
28^36 
28^60 
28^72 
Base 42:
42^20 
42^32 
42^44 
42^56 
42^80 
Base 50:
50^60 
50^72 
50^90 
Base 98:
98^36 
98^60 
98^72 
Base 200:
200^36 
200^60 
Base 220:
220^24 
220^54 
Base 242:
242^18 
242^36 
Base 284:
284^30 
284^60 
Base 338:
338^24 
338^36 
338^48 
Base 392:
392^36 
392^48 
Base 496:
496^30 
496^36 
Base 578:
578^12 
578^16 
578^24 
578^36 
578^40 
578^48 
Base 882:
882^40 
Base 1155:
1155^11 
1155^23 
1155^29 
1155^35 
1155^47 
Base 2310:
2310^14 
Base 8128:
8128^36 
Base 30030:
30030^14 
Base 510510:
510510^14 
Base 9699690:
9699690^14 
Base 223092870:
223092870^14

I did a few checks, but not on the entire list.

[EdH]

Originally Posted by garambois

. . .
But for the sequence 6^188, this amounts to factoring a C146, which of course poses more problems !
. . .

I ran a YAFU Colab instance against the C146 and it knocked down to a C104. If no one else claims it, I'll factor the C104 tomorrow.

Edit: I just realized that yoyo has these reserved, but the initial term is greater than 139 digits!

@yoyo: Should I leave the C104 for you, or go ahead and factor it? The index is at 147 digits. The other sequence was at 143 digits, but the cofactor was C115. Sorry if I overstepped your reservation.

[yoyo]

No problem, you can factor it.

[garambois]

Originally Posted by EdH

Let me make sure I have this right:

You'd like a list showing any matched parity sequences with index 2 > index 1?

As an example, 2^12:

0 .   4096 = 2^12 
1 .   4095 = 3^2 * 5 * 7 * 13 
2 .   4641 = 3 * 7 * 13 * 17 
3 .   3423 = 3 * 7 * 163 
4 .   1825 = 5^2 * 73 
5 .   469 = 7 * 67 
6 .   75 = 3 * 5^2 
7 .   49 = 7^2 
8 .   8 = 2^3 
9 .   7 = 7

I'm working on a program to find all such sequences within the tables. Am I on the right track?

This is exactly it.
And if you write the program, you will see the conjecture that I will present tomorrow.
I draw your attention to the bases which are primorial numbers and primorial numbers without the factor 2.

[garambois]

Originally Posted by EdH

Base 6, 12 and 30 so far, don't appear to have any sequences with index 2 > index 1, but base 15 has 15^35 and many others exist throughout the bases. Here's a list of all I found in the current tables:
...
...

I did a few checks, but not on the entire list.

Now you have an idea of the conjectures I was able to formulate : try to guess them ?
For example, do you see exponent 14 for the primorial bases ?
And above all, you are missing the odd bases (with odd exponents of course) that do not appear in your table !
The primorials without the factor 2 are very special compared to the other odd numbers, I won't tell you more, we'll see if you come to the same conclusions as me if you try it out.
It was when I started to observe this kind of thing 15 days ago that I asked you to complete sequences or add some bases.
Tonight or tomorrow morning, I will scan all of the project bases taking into account the new data.

[garambois]

Originally Posted by EdH

Edit: I just realized that yoyo has these reserved, but the initial term is greater than 139 digits!

@yoyo: Should I leave the C104 for you, or go ahead and factor it? The index is at 147 digits. The other sequence was at 143 digits, but the cofactor was C115. Sorry if I overstepped your reservation.

I'm sorry, this is my fault, I should have reminded in my post # 892 that these bases had been reserved by yoyo.
I must admit that in the haste, I did not think about this any more.
I thought to myself that I needed to have index 2 of as many of these sequences as possible very quickly before tomorrow morning (March 2), for when I will do the full scan of all the databases.
I am really sorry !