mersenneforum.org Aliquot sequences that start on the integer powers n^i
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 2021-07-29, 22:57 #1255 richs     "Rich" Aug 2002 Benicia, California 142010 Posts 392^60 terminates P69 at i99.
 2021-07-31, 11:55 #1256 richs     "Rich" Aug 2002 Benicia, California 26148 Posts 392^53 terminates P15 at i80. Last fiddled with by richs on 2021-07-31 at 11:55 Reason: Typo
 2021-07-31, 12:04 #1257 Happy5214     "Alexander" Nov 2008 The Alamo City 19·41 Posts 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.
 2021-08-01, 13:36 #1258 Happy5214     "Alexander" Nov 2008 The Alamo City 77910 Posts 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.
 2021-08-01, 14:48 #1259 Happy5214     "Alexander" Nov 2008 The Alamo City 19×41 Posts 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. Last fiddled with by Happy5214 on 2021-08-01 at 14:48 Reason: Note 5-cycle
 2021-08-01, 16:22 #1260 RichD     Sep 2008 Kansas 1101101000002 Posts Terminates: 722^51 722^53 Begin initializing base 51.
 2021-08-01, 17:40 #1261 yoyo     Oct 2006 Berlin, Germany 23·79 Posts Take bases 578 722 770 882.
 2021-08-02, 03:28 #1262 sweety439     "99(4^34019)99 palind" Nov 2016 (P^81993)SZ base 36 C8E16 Posts 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.
 2021-08-02, 05:06 #1263 Happy5214     "Alexander" Nov 2008 The Alamo City 19·41 Posts 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.
 2021-08-02, 08:55 #1264 garambois     "Garambois Jean-Luc" Oct 2011 France 2×3×7×17 Posts 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.
 2021-08-02, 09:36 #1265 sweety439     "99(4^34019)99 palind" Nov 2016 (P^81993)SZ base 36 2·1,607 Posts 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. Last fiddled with by sweety439 on 2021-08-02 at 09:36

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