Aliquot sequences that start on the integer powers n^i

[richs]

392^60 terminates P69 at i99.

[richs]

392^53 terminates P15 at i80.

[Happy5214]

I'm done initializing 14264, and 14536 will be done by late Saturday. 15472 will be finished no later than Monday, and there will be plenty to say about that last base.

[Happy5214]

14536 is done. I'm down to one more sequence for 15472, which has hit a downdriver. I may have to interrupt it to run some other tasks, but it should be done by tomorrow.

[Happy5214]

OK, 15472 is initialized, and it is a gold mine. The initialization found 2 particularly noteworthy sequences, i=17 and 22, both terminating at different perfect numbers. Note that one is non-trivial. The 5-cycle is completely initialized.

[RichD]

Terminates:
722^51
722^53

Begin initializing base 51.

[yoyo]

Take bases 578 722 770 882.

[sweety439]

276 is interesting since it is the smallest number whose aliquot sequence has not yet been fully determined.

However, 276^2 immediately terminated at the prime 146683.

276^3 currently at 93-digit number with C92

276^4 terminated at 43

276^5 has 1140 steps to get a 83-digit number with C80

276^6 terminated at 109

276^7 currently at 82-digit number

276^8 terminated at a 20-digit prime after 2 steps

276^9 currently at 81-digit number

276^10 terminated at 19

276^11 merges with 25911768

276^12 terminated at a 14-digit prime after 7 steps

Conjectures:

* If n is odd, then 276^n never terminate.
* If n is even, then 276^n must terminate.

[Happy5214]

Reserving all of the Lehmer five to initialize, with the proviso that I may not get to all of them. I'll try to promptly release any I know I won't end up getting to.

IMO bases which themselves are main project sequences (like the Lehmer five) should form a new category on the main page given their particular notability.

[garambois]

Hello everyone, I'm back from vacation.
Meije is really a marvelous mountain and all the massif of Écrins in general !

Thanks to all for the many works done since two weeks.
I will take into account all your messages.
I will start by updating the page completely, adding all your new initialized sequences, all your reservations and everything else.
It might take me two or three days with the checks given all the new stuff.
Then I'll start analyzing the data.
I don't know how long this will take, as I think the amount of data has increased at least tenfold since last year.
And we'll see if it leads to new and interesting remarks.
I'm going to focus my attention on sequences that end on cycles, hoping that we'll have enough data to try to notice something.
And of course, I'm going to look closely at the prime numbers that end sequences according to the bases and base categories.
I'll also "randomly poke around" in the data to try to see some totally unexpected things.
I'll keep you posted.

[sweety439]

The page has "Primorials", but does not have "Factorials", I try to take the factorials.

Also, I try the highly abundant numbers, since they are the numbers whose sigma function sets record, and sigma function is highly related to Aliquot sequences.

Besides, there are also interesting bases: 102 and 138, see https://oeis.org/A098009, they set record for the length of Aliquot sequences.

Finally, not only the Lehmer five, there are also other numbers less than 1000 which is conjectured to have an infinite, aperiodic, aliquot sequence: 306, 396, 696, 780, 828, 888, 996, which have the same trajectories as the Lehmer five.