mersenneforum.org Aliquot sequences that start on the integer powers n^i
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 2018-08-28, 13:27 #1 garambois     "Garambois Jean-Luc" Oct 2011 France 809 Posts Aliquot sequences that start on the integer powers n^i On my website, I wrote a page that summarizes my work on aliquot sequences starting on integer powers n^i. This page summarizes the results and reservations for each aliquot sequences one has chosen to calculate. See this page. If someone in this forum also wants to calculate these aliquot sequences with me, he can indicate it to me here and I note his name in the cells of my page to reserve him the integer powers of his choice. He will then have to enter the results into factordb and let me know so that I can fill and color the cells of the array as appropriate. Note : For open-end aliquot sequences, I stop at 10^120 (orange color cells). Edit: The following link will take you to a regularly updated page of conjectures that have been formed based on this project: Conjectures related to aliquot sequences starting on integer powers n^i Last fiddled with by EdH on 2021-04-03 at 15:59 Reason: Added new link to Conjectures page.
 2018-08-29, 15:38 #2 LaurV Romulan Interpreter     "name field" Jun 2011 Thailand 35·41 Posts I am (occasionally) working on 6^n, the progress for the larger seq is due to me, in the past.
 2018-08-30, 12:52 #3 garambois     "Garambois Jean-Luc" Oct 2011 France 32916 Posts OK LaurV, Do you remember for which i values you calculated the aliquot sequences of 6^i ? So I can add your name instead of "A" (Anonymous) in the array.
 2018-08-30, 17:56 #4 LaurV Romulan Interpreter     "name field" Jun 2011 Thailand 35·41 Posts Below 20 (and inclusive) the only sequences which do not terminate are 6^(7, 9, 11, 15, 19). They are all at the point where I left them, except for few terms of 6^15 added by the elves where the last cofactor was under 110 digits. I work on 10077696 (6^9), and I reserved 95280 years ago, when 279936 (6^7) merged with it. I brought it to 148 digits (currently with a C140 cofactor, I still have it reserved, but not in priority list due to the 2^4*31 driver). Interesting that even powers (including higher, to 30 or 50, can't remember) all terminate, some in very large primes. Also, odd powers between 21 and 31 were left after C>100, and were advanced a bit by the DB elves. Last fiddled with by LaurV on 2018-08-30 at 17:56
2018-08-30, 21:42   #5
garambois

"Garambois Jean-Luc"
Oct 2011
France

809 Posts

Quote:
 Originally Posted by LaurV Below 20 (and inclusive) the only sequences which do not terminate are 6^(7, 9, 11, 15, 19). They are all at the point where I left them, except for few terms of 6^15 added by the elves where the last cofactor was under 110 digits. I work on 10077696 (6^9), and I reserved 95280 years ago, when 279936 (6^7) merged with it. I brought it to 148 digits (currently with a C140 cofactor, I still have it reserved, but not in priority list due to the 2^4*31 driver).
OK, I wrote your name on the page.

Quote:
 Originally Posted by LaurV Interesting that even powers (including higher, to 30 or 50, can't remember) all terminate, some in very large primes. Also, odd powers between 21 and 31 were left after C>100, and were advanced a bit by the DB elves.
Yes, If n is even and if and only if n takes the form n=m^2 or n=2*m^2, then sigma(n)-n will be odd.
If n is odd and if and only if n takes the form n=m^2, then sigma(n)-n will be even.

See the proof, but sorry, in french.

So, because n=6^i always is even, and n takes the form n=m^2 only when i is even, then, sigma(n)-n will be odd and the aliquot sequence will go down. If i is odd, then we will probabily have an Open-End aliquot sequence.

If for example, we take n=5^i which always is odd, when i is even then n takes the form n=m^2, so sigma(n)-n will be even, so the aliquot sequence will probabily be Open-End.

If for example, we take n=2^i which always takes the form m^2 (if i is even) or the form 2 * m^2 (if i is odd), then, sigma(n)-n will always be odd, so the aliquot sequence will probabily always go down.

I would like to find one i with 2^i an open-end sequence, but I haven't found such an aliquot sequence yet.

 2018-09-19, 19:40 #6 garambois     "Garambois Jean-Luc" Oct 2011 France 11001010012 Posts Thanks to Karsten Bonath for completely redesigning my calculation tracking web page. The new page is much more readable and allows immediate access to the data on FactorDB with a simple click. For the moment, only some aliquots sequences for n=2^i, 3^i, 6^i and 11^i are reserved.
 2018-09-19, 23:22 #7 RichD     Sep 2008 Kansas 3,571 Posts AS 11^56 terminates. Edit: AS 11^58 terminates. Last fiddled with by RichD on 2018-09-19 at 23:46
 2018-09-20, 01:19 #8 RichD     Sep 2008 Kansas DF316 Posts AS 10^108 terminates. AS 10^104 terminates, not by me.
 2018-09-21, 00:03 #9 wpolly     Sep 2002 Vienna, Austria 21910 Posts 6^77 terminates.
 2018-09-23, 20:26 #10 garambois     "Garambois Jean-Luc" Oct 2011 France 11001010012 Posts OK, RichD and wpolly, page updated. Thank you for your help !
 2018-09-25, 14:04 #11 RichD     Sep 2008 Kansas 1101111100112 Posts 10^106 terminates with a prime - 91909. Last fiddled with by RichD on 2018-09-25 at 14:05 Reason: 10, not 11

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