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Old 2011-12-14, 13:50   #1
Drdmitry's Avatar
Nov 2011

23·3·11 Posts
Default Small search of cycles with odd and even elements

I have made small search of aliquot cycles which contain both even and odd elements. I hope it will be of interest in this forum.

Surely such cycles should contain at least one element of the form x = n^2 where n is odd and at least one element of the form 2^d * n^2 where d>=1 and n is odd.

All the sequences starting from elements below 4 000 000 were tested upto the limit of 70 digits. (According to empirical evidence from these limits were pushed much further however I have not found any exact evidence of that). So if the element of aliquot sequence becomes smaller than 4 millions then I just give it up.

Firstly I tested all sequences starting from odd squares between 4 * 10^6 and 16 * 10^6. There are 1000 candidates to test. At the beginning I tested them with my own programme until sequence reaches 16 digits length. Only 96 sequences survive this procedure. Then I tested them by Aliqueit until they push the limit of 70 digits. Finally i've got 80 survivors.

Secondly I tested all sequences starting from even number of the form 2^d * n^2 smaller than 10^14. One can calculate that in total there are 12071067 candidates to test. In this case I give the number up if one of two things happen:
- element of the sequence becomes smaller than 4 millions.
- if the switch odd -> even happens with odd element smaller than 16 millions (this case has already been searched before).
This time only 377 candidates reached 16 digits limit. All of them except one sequence switched the parity of their elements twice (even -> odd -> even). Finally 266 of them reached the limit of 70 digits.

None of the cycles with odd and even elements had been found. So we can state that if such cycle contains number 2^d * n^2 < 10^14 then the largerst element of this cycle is at least 10^69.
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