Nonaveraging sets search

A nonaveraging set is a set with no 3 terms in an arithmetic progression. For a given n let a(n) be the smallest number so that n-element nonaveraging set can be selected from 1,2,3,...,a(n).
I am also interested in modular solutions as they are of great use to construct integer solutions. A modular solution is an n-element nonaveraging set mod m(n), where we want to minimize m(n). Note that if m(n) is even, then terms i and i+m(n)/2 exclude each other.
The main goal is to prove as much as possible about NrootN conjecture.
Search results: Solutions submitted Integer records Modular records

Submitting solutions:

To submit a solution send email to jwr@math.uni.wroc.pl. In the body of the email include a line with exactly 37 hash (#) signs, then line with your name (less than 80 characters), and then your solutions.
Each solution should be a sequence of integers separated by spaces or newlines. It should contain nonnegative integers below 60000 ordered in increasing order. The length of each solution should be between 3 and 1024.
Integer solution cannot contain zero term. It should have 0 at the end to indicate the end of the solution.
Modular solution can contain zero term, but must have all terms less than modulus. It should have minus modulus at the end to indicate the end of the solution.
At the end include number -1 to indicate there are no more solutions. Note that in case of errors or invalid solutions, all remaining solutions will be ignored.
For example your mail could look like that:
``````
#####################################
Jimmy Whatever
1 2 6 7 9 14 15 18 20 0
1 2 5 7 11 16 18 19 23 24 0
1 2 5 7 11 16 18 19 23 24 26 0
1 3 4 8 9 11 20 22 23 27 28 30 0
1 2 4 8 9 11 19 22 23 26 28 31 32 0
0 1 3 12 14 15 -19
0 1 3 4 10 12 22 26 28 -35
0 1 3 7 17 24 25 28 29 35 -37
0 1 3 7 8 10 18 32 37 38 41 43 -52
0 1 3 4 9 11 29 39 42 43 46 48 51 -61
-1
``````
It contains 5 integer solutions and 5 modular solutions, mod 19, 35, 37, 52 and 61.
I will forward your submission into a verification program and will mail the log of verification process back to you. This will be done by hand, so be patient, it may take even a few weeks if I happen to be away. If you set a new record, your solution will appear here. A strange number in parentheses is time of your submission in seconds elapsed since 1980 or whatever. I do not have time to play with that right now, but in future I should use it to give proper credits.
If you set a new record, there will be an update of integer and modular records. It will include an update of record you have set and all results implied by it using methods known to the program. Method will appear next to the result and I will proceed to include expalanation somewhere on this website. For modular solutions Log[n,m(n)] is also included as a measurement of their quality.
In case of a submission as above, log of the verification session could look like that
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#       #   ####   #       #       #     #       #  ########
##      #  #    #  ##      #      # #    #       #  #
# #     # #      # # #     #      # #     #     #   #
#  #    # #      # #  #    #     #   #    #     #   #
#   #   # #      # #   #   #     #   #     #   #    #####
#    #  # #      # #    #  #    #######    #   #    #
#     # # #      # #     # #    #     #     # #     #
#      ##  #    #  #      ##   #       #    # #     #
#       #   ####   #       #   #       #     #      ########

Dear Jimmy Whatever !
Thank you for submitting nonaveraging sequences.

Solution 1 has 9 terms and maximum 20 (best known 20)
Solution 2 has 10 terms and maximum 24 (best known 24)
Solution 3 has 11 terms and maximum 26 (best known 26)
Solution 4 has 12 terms and maximum 30 (best known 30)
Solution 5 has 13 terms and maximum 32 (best known 32)
Solution 6 has 6 terms and modulus 19 (best known 19)
Solution 7 has 9 terms and modulus 35 (best known 35)
Solution 8 has 10 terms and modulus 37 (best known 37)
Solution 9 has 12 terms and modulus 52 (best known 52)
Solution 10 has 13 terms and modulus 61 (best known 61)
No more solutions.

Jarek

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Note that if you see
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Solution 7 has 9 terms and modulus 35 (best known 35)
``````
that means solution 7 has been verified. All error messages just after that apply to solution 8.
Jarek Wroblewski jwr@math.uni.wroc.pl