Aliquot sequences that start on the integer powers n^i

[garambois]

Page updated !
All the bases have been updated !
I only had problems with base 29, but it's all fixed now.
Bases 72, 220 and 284 added.
There is no abnormal behavior for bases 220 and 284. But we still had to check.

Thank you all for your precious help !
I also thank you to check if nothing seems abnormal for the bases you have been working on after this big update.

Since last summer, we have accumulated a very large amount of new data. And it's still going on.

[EdH]

The last two base 2 table holdouts have been convinced to turn green!

[garambois]

Okay, thank you very much Edwin.
The table of base 2 will be all green at the next update... And I love green !

[garambois]

Page updated.
Many thanks to all for your help !

The next update will take place in 2 or 3 weeks, depending on the number of requests that will be posted here.
Because now, the calculations will progress more slowly, since the sequences have all arrived at large terms.

[VBCurtis]

How unusual is it to break 2^4 * 31? I don't think I've done it before tonight.

Edit: The sequence went 2^4 * 3 * 31 to 2^4 * 3 * 31^2 to 2^6 * 3 * 31, and picked up the downdriver a few terms later.

[garambois]

Please, which sequence ?

[VBCurtis]

13^64. It got down under 50 digits, but picked up 2 * 3 so it's back above 110 dig now.

13^58, 60, 62 can be updated on your page also.

[EdH]

Originally Posted by VBCurtis

How unusual is it to break 2^4 * 31? I don't think I've done it before tonight.

Edit: The sequence went 2^4 * 3 * 31 to 2^4 * 3 * 31^2 to 2^6 * 3 * 31, and picked up the downdriver a few terms later.

I scanned all the sequences for all the tables for the project page and came up with the following:

For 957 sequences that contain at least one 2ⁿ * 31 run, no matter the length, 74 of these sequences still have 2ⁿ * 31 within their last term, which leaves 883 which have broken it. I used values of 4 through 16 for n and this power may have changed within the run. I did not look for exponents for 31. If a sequence had more than one run of 2ⁿ * 31, only one was counted.

[VBCurtis]

That's not unusual, at all! I guess I just don't encounter 2^4 * 31 often enough to pay attention. I appreciate the data!

[garambois]

Originally Posted by VBCurtis

13^64. It got down under 50 digits, but picked up 2 * 3 so it's back above 110 dig now.

13^58, 60, 62 can be updated on your page also.

Ok for 13^64, thanks...

The next update will take place next weekend.

[EdH]

Originally Posted by VBCurtis

How unusual is it to break 2^4 * 31? I don't think I've done it before tonight.

Edit: The sequence went 2^4 * 3 * 31 to 2^4 * 3 * 31^2 to 2^6 * 3 * 31, and picked up the downdriver a few terms later.

My earlier numbers are probably quite inaccurate! I may need to focus on 2^4 only, rather than 2^(>4). I did a bit more study and it appears that 2⁴ * 31 is specifically important! If the power of either 2 or 31 changes, the driver has a really good chance of breaking. There are instances of 2^(>4) * 31 and 2^4 * 31² not starting a run.

Example for 10^27 (first column is index number):

52 = 2^7 * 31 * 37 . . .
170 = 2^5 * 3^2 * 31 * 41 . . .
687 = 2^6 * 3 * 31^2 * 59 . . .
754 = 2^4 * 7 * 31 * 223 . . .
755 = 2^4 * 7 * 31 * . . . 
. . . (2^4 continues for all these lines)
1081 = 2^4 * 3 * 13 * 31 * 2207 . . .
1082 = 2^4 * 3 * 5 * 31 * 47681 . . .

1082 is the current last term, so this instance is unbroken.


Another example 10^111:

592 = 2^4 * 31 * 73 . . .
593 = 2^4 * 31 * 1493 . . .
. . .
674 = 2^4 * 13 * 31 * 128311 . . .
675 = 2^4 * 5 * 31^2 * 1407503 . . .
676 = 2^4 * 5^2 * 31^2 * 89 . . .
677 = 2^4 * 5^2 * 17 * 31^2 . . .

1087 is the current last term, so this instance broke at 677.

I will do even more study and post something extra later. . .