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Old 2021-06-27, 19:10   #1200
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Ok thanks. 53^30 terminated so it definitely merges with some lower sequence and a few others went below 15 digits. But I'll continue finishing the sequences to 110 digits and then check for merges in one go. Might waste some computation time on select occasions, but is easier for me.
Sounds good. Unless you've had a substantial downswing, a merge is not likely.
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Old 2021-06-28, 12:34   #1201
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I used your software, it detected the merge of open ended 53^10, but not for the terminating 53^30. Is it supposed to be like that?

Btw, does factordb detect merges? That would be quite a useful feature.
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Old 2021-06-28, 12:55   #1202
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I used your software, it detected the merge of open ended 53^10, but not for the terminating 53^30. Is it supposed to be like that?

Btw, does factordb detect merges? That would be quite a useful feature.
Only merges that are still open-ended matter for our purposes. Anything that terminates, merges with a whole family of sequences - all the ones that terminate with that value.

factordb only works directly with composites and factors. It does track merges in a manner, but that info is only available if you look for it. On a page for the first value that exists in both merged sequences, look at the "More information" section. Under the "Others:" block you will see previous terms that lead to the current one. If you follow those back up their sequences, you will arrive at the source sequences. I have used that method to find merges outside the 3*10^6 listing of C80s.
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Old 2021-06-28, 18:18   #1203
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I tried going along the "More information" path, but figured it might take very long with uncertain outcome.


I noticed that for prime b: s(b^n) = (b^n-1)/(b-1). It's probably well known and even I realized it's simply because s(b^n) = 1 + b^1 + b^2 + ... b^(n-2) + b^(n-1) is a geometric series, but I was wondering if that doesn't mean that if b and n satisfy b^n = (b^n - 1) / (b-1), then b^n is a perfect number?


I'm not sure though if that equation even has solutions.

Last fiddled with by bur on 2021-06-28 at 18:19
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Old 2021-06-29, 00:31   #1204
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I tried going along the "More information" path, but figured it might take very long with uncertain outcome.


I noticed that for prime b: s(b^n) = (b^n-1)/(b-1). It's probably well known and even I realized it's simply because s(b^n) = 1 + b^1 + b^2 + ... b^(n-2) + b^(n-1) is a geometric series, but I was wondering if that doesn't mean that if b and n satisfy b^n = (b^n - 1) / (b-1), then b^n is a perfect number?


I'm not sure though if that equation even has solutions.
For \(b \neq 1\), \(b^n = (b^n - 1) / (b-1) \Leftrightarrow b^n(b-1) = b^{n+1}-b^n = b^n-1 \Leftrightarrow b^{n+1}-2b^n+1 = 0\). I plugged that into Wolfram Alpha, and it gave me the solution formula of \(n = -\frac{\log(2-b)}{\log(b)}\), which wouldn't appear to allow useful integer solutions for both b and n.
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Old 2021-06-30, 12:39   #1205
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Happy, too bad, but it'd been a surprise if there was an integer solution.

Quick update on the 53 base, I ran all n <= 100 to a co-factor of at least 105 digits. Now I'm trying to terminate all the trivial sequences that were too large for factordb to take care of them, starting with n = 69. There are quite a few co-factors larger than 150 digits, so it might take some time.
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Old 2021-06-30, 14:13   #1206
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Now I'm trying to terminate all the trivial sequences that were too large for factordb to take care of them...
This may be feasible, as there is no factor 3 for these sequences.
But indeed, it should take some time.
Thank you very much for all this work !
I will add the 53 base to the next update.
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Old 2021-06-30, 19:16   #1207
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Yes, they decrease relatively fast, there was even a semiprime once, that caused a tremendous drop in size.



But no need to thank me, I could as well thank you for organizing it. And if some of the larger ones really withstand ECM they could at least be used to help vbcurtis optimize his cado parameters.
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Old 2021-07-01, 17:38   #1208
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This may be feasible, as there is no factor 3 for these sequences.
Actually there are sequences with factor 3, for example 53^77 and others. Is there a rule when that factor should appear?
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Old 2021-07-01, 23:12   #1209
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Actually there are sequences with factor 3, for example 53^77 and others. Is there a rule when that factor should appear?
There's not an obvious rule for when it appears in the middle of a sequence. We can often tell when it will appear at the start though, especially for a prime base.

---------

Base 48 is done. I got to i=80 on the trivial sequences (135 digits). Base 24 is also done to i=50. If nobody reserves the priority cycle bases by the time I finish my catch-up with GIMPS, I may take one or both of those.
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Old 2021-07-02, 17:27   #1210
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Base 48 is done. I got to i=80 on the trivial sequences (135 digits). Base 24 is also done to i=50. If nobody reserves the priority cycle bases by the time I finish my catch-up with GIMPS, I may take one or both of those.
OK, many thanks for your help.
I will add the base 48 in the next update.
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