Aliquot sequences that start on the integer powers n^i

[EdH]

Originally Posted by Happy5214

Not conjecture, theorem. If p = ab (a, b > 1), then 2^a-1 and 2^b-1 both divide 2^p-1. Ergo, any number that divides 2^n-1 will also divide 2^(ni)-1, for any i ≥ 1. That's why exponents for Mersenne primes must themselves also be prime.

Does this apparent observation fit in with a similar theorem?

For all aⁱ (a, i positive integers ≥ 1)
s(aⁱ) is a factor of s(a^(i*n)) (for all positive n)

Example:

s(73) = 3 · 19
s(7(3*2)) = 2^3 · 3 · 19 · 43
s(7(3*3)) = 3^2 · 19 · 37 · 1063
. . .
s(7(3*33)) = 3^2 · 19 · 37 · 199 · 1063 · 1123 · 3631 · 173647 · 293459 · 1532917 · 12323587 · P44
. . .

Note also from the above:

s(7(3*3)) = 3^2 · 19 · 37 · 1063
. . .
s(7(3*33)) = 3^2 · 19 · 37 · 199 · 1063 · 1123 · 3631 · 173647 · 293459 · 1532917 · 12323587 · P44

Edit: Further study seems to suggest the above is only true for odd a. Additionally, that aⁱ+1 is a factor of s(a^(i*n)) (n, a positive even integer)

[unconnected]

If n=13 available for reservation I'd like to reserve range from l=80 to 120 digits.

[VBCurtis]

Originally Posted by unconnected

If n=13 available for reservation I'd like to reserve range from l=80 to 120 digits.

Well, I've been working in spurts on n=13 for a couple of years. I have reserved only up to 13^60, but I do plan to cover all of it and I've just 2 sequences left to begin in my reservation to 13^60.

I'm not sure what you mean by "from 80 to 120 digits".... is that the starting size of the sequences, or that you want to take all remaining sequences to 120 digits that I haven't reserved? If you mean the latter, how about we split the rest of the sequences- I'll take up to 13^78, you take 13^80 and up?

[unconnected]

I mean that I'll take all sequences from 13^80 to 13^106 and promote them to at least 120 digits. If this is OK for you then I'll start.

[VBCurtis]

Originally Posted by unconnected

I mean that I'll take all sequences from 13^80 to 13^106 and promote them to at least 120 digits. If this is OK for you then I'll start.

Excellent! Be my guest.

Also, I'd like to reserve 13^60 up to 13^78. I'll start them next week.

[richs]

Originally Posted by richs

Reserving 439^24.

439^24 is now at i1448 (added over 1300 iterations with a good downdriver along the way) and a C121 level with a 2^2 * 3 * 5 * 7 guide, so I will drop this reservation. The remaining C118 term is well ecm'ed and is ready for NFS.

Reserving 439^26 at i373.

[Happy5214]

I've finished n=21 up to i=70, and I'll release those sequences. Right now, I'm going to fill in the first row of n=24 (only 3 sequences left).

[EdH]

@Jean-Luc: My version of primes>1 listings are attached below for all the tables currently on the page. Although my listing has a different format from yours, the details in my base2primes listing contain those for your listing and they all appear to match. I've included all primes that show up more than once within a base, even the smaller ones. I haven't done a check for matching primes across bases.

Here's a brief example of my format compared to yours:

base2primes:

. . .
prime 197748738449921 shows up 2 times (265, 530).
prime 242099935645987 shows up 2 times (198, 396).
prime 332584516519201 shows up 2 times (191, 382).
. . .

. . .
base 2    prime 197748738449921    exponent 265
base 2    prime 197748738449921    exponent 530

base 2    prime 242099935645987    exponent 198
base 2    prime 242099935645987    exponent 396

base 2    prime 332584516519201    exponent 191
base 2    prime 332584516519201    exponent 382
. . .

[EdH]

I went ahead and did all the preliminary work for base 30030. There are two merges:

30030^1:i1 merges with 22518:i4
30030^19:i841 merges with 41364:i4

All the opens are at least 100 dd and the rest are terminated with primes. Leave it unreserved for now. Someone else can have it, if they want. I'm not sure if I'll take the opens to 120 dd later, or not.

Edit: I have decided to go ahead and turn all the transparent cells to a shade of orange. (I've crossed out 30030 in post 280.)

[EdH]

Base 30030 is all colored in and I've attached the list of primes that appear more than once.

[EdH]

Row 520-539 for base 2 has completely turned green (all run down to primes).

I am currently doing all the preliminary work for a table to be added for 2310. I'm not sure if I will color in the transparent cells or not (like before with 30030).