Aliquot sequences that start on the integer powers n^i

[garambois]

Originally Posted by yoyo

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

OK, thanks a lot !
At the next update, I will reserve the base 12 for you.
No, you don't take too much !
It all depends on YAFU's computing means.
YAFU's statistics are very high at the moment.
I don't know if a team is raiding on YAFU and if the statistics will stay so high in the next few times ?

[garambois]

Originally Posted by EdH

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).

Ok, this time the file is perfect !
Thanks a lot Ed.

I examined it closely.
Unfortunately, nothing special catches my attention at the moment.
Except a detail for the prime number 53 : there are only sequences that start on powers of 2 that end with the prime number 53. But this must be pure chance !

But maybe someone else will observe something interesting...

[garambois]

I would also like to make a comment about a private conversation that Edwin Hall and I had in early July. Edwin allowed me to talk about this private conversation here when the messages are readable by everyone.
Here is Edwin's observation :

Sequences that had abundant indices somewhere and a parity change other than at index 1 (all were due to perfect squares):
(2^9, 2^62, 2^210): 81 >> 40
(2^12, 2^141, 2^278, 2^387):  49 >> 8
(2^112): 2209 >> 48
(2^117): 25921 >> 5600
(2^141): (729 >> 364) and (49 >> 8)
(2^243): 1225 >> 542
(2^271): 2025 >> 1726
(2^305, 2^317): 9 >> 4
(2^421): 169 >> 14
Of the even numbers that were due to parity flips, only 40, 48, 364 and 5600, are abundant.

This means, for example, that sequences that begin with 2^9, 2^62 and 2^210 arrive on the integer 81, which is a perfect square and therefore, at the next iteration, we will have an even term (here, 40, because s(81)=40). The goal was to find parts of sequences that begin with powers of 2 and that are increasing. The terms are then abundant.

I think we can generalize this study by looking at the table called "base_2_mat" that I attached to post #447.
Here is an excerpt from this table :

prime 2 in sequence 2^1 at index 0
prime 2 in sequence 2^2 at index 0
prime 2 in sequence 2^3 at index 0
prime 2 in sequence 2^4 at index 0 3
...
prime 2 in sequence 2^9 at index 0 3 4
prime 2 in sequence 2^10 at index 0 5
...
prime 2 in sequence 2^12 at index 0 8
...
prime 2 in sequence 2^14 at index 0 7 8
prime 2 in sequence 2^15 at index 0 7 8
...
prime 2 in sequence 2^55 at index 0 14
...
prime 2 in sequence 2^59 at index 0 12 13
...
prime 2 in sequence 2^62 at index 0 26 27
...
prime 2 in sequence 2^112 at index 0 62 63 64
...
prime 2 in sequence 2^117 at index 0 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56
...
prime 2 in sequence 2^141 at index 0 25 26 27 28 34
...
prime 2 in sequence 2^164 at index 0 42
...
prime 2 in sequence 2^210 at index 0 44 45 46 49 50
...
prime 2 in sequence 2^243 at index 0 128 129 130 131
...
prime 2 in sequence 2^271 at index 0 79 80 81 82 83 84 85 86 87 88 89
...
prime 2 in sequence 2^278 at index 0 51
...
prime 2 in sequence 2^305 at index 0 76
...
prime 2 in sequence 2^317 at index 0 70
...
prime 2 in sequence 2^373 at index 0 94 95 96 97
...
prime 2 in sequence 2^387 at index 0 102
...
prime 2 in sequence 2^421 at index 0 65 66 67
...
prime 2 in sequence 2^510 at index 0 125 126
  ...

For more readability, I removed all the lines where there was only the index 0 of concerned. There are dotted lines instead. When you see this table, you can immediately see in which sequences from base 2, the prime number 2 appears in the factorization of terms at an index other than 0. All sequences with increasing parts identified by Edwin are found in this way, plus others: those with even terms but which are in "downdriver".


Unfortunately, here again: I don't notice anything about the exponents, nor the indexes that could allow to predict for which base 2 exponents, we have parts of sequences with the prime number 2 elsewhere than at the index 0 !!!

[RichD]

I'll start work on Table n=29.

[RichD]

This may be of interest because it flips parity.

http://factordb.com/sequences.php?se...ge&fr=32&to=42

[Sergiosi]

Reserving 2310^25

[garambois]

@RichD :
OK for base 29.
OK for 265^2, thank you very much ! This number was already in my tables...


@Sergiosi :
I think you have already finished the calculations for 2310^25. And moreover, it is a non-trivial end, which is rare ! Thanks a lot !

[RichD]

Who has the way back merge detection meter? I believe 20^37 has merged.

[EdH]

Originally Posted by RichD

Who has the way back merge detection meter? I believe 20^37 has merged.

20^37:i1855 merges with 660:i25

[richs]

Originally Posted by richs

Reserving 439^30 at i80.

439^30 is now at i124 (added 44 iterations) and a C121 level with a 2^6 * 3 guide, so I will drop this reservation. The remaining C119 term is well ecm'ed and is ready for nfs.

Taking 439^36 at i68.

[RichD]

19^14 might have also merged.