# Nonaveraging sets - what's new

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).
Click here for search details and results.
For n<=41 the exact values of a(n) are known, see The On-Line Encyclopedia of Integer Sequences.
October 16, 2008
Rodolfo Niborski verified that a(41)=194. It took him 11.5 days on a 2.8 GHz CPU.
August 13, 2008
a(45)<=227 from Clyde Kruskal. Congratulations !!!
December 1, 2007
More improvements from Gavin Theobald!
Also he has confirmed that a(37)=163, a(38)=165, a(39)=169 and a(40)=174.
He conjectures that a(41)=194, but it would take him well over a year to verify that.
June 24, 2005
More improvements from Gavin Theobald!
September 20, 2004
Using Gavin Theobald's great results I was able to verify NrootN conjecture for n<=1,408,000 !!!!
September 15, 2004
Great improvements from Gavin Theobald! Now we know

# a(128)<=1092

(best estimate known before 1094). Also m(40)<=275 beats my m(216)<=3889 !!! Congratulations !
Now I am very busy, but next week I will see what is the status of NrootN conjecture now.
September 6, 2004
New improvements from Gavin Theobald! Now we know

# a(256)<=3208

(best estimate known before 3251). m(31)<=202 looks impresive too. Congratulations !
August 29, 2004
New improvements from Gavin Theobald! Congratulations !
August 18, 2004
I had a careful look at Gavin Theobald's results. All results marked with "Theorem 24" are now derived from his m(24)<=148.
It also allowed to verify NrootN conjecture up to 40,000 and up to 50,000 with a small gap, as can be seen here.
July 21-22, 2004
New improvements from Gavin Theobald! Congratulations !
July 12, 2004
Gavin Theobald confirmed a(36)=157. This took about 16 days of processing!
July 4, 2004
New improvements from Gavin Theobald! Congratulations !
June 21, 2004
Gavin Theobald again! 8 new improvements! Congratulations !
June 20, 2004
Gavin Theobald confirmed a(35)=150 by an exhaustive search (5 days of a 1.4 GHz processor work).
June 14, 2004
Gavin Theobald submitted a lot of new improvements since last update. Congratulations !
May 27, 2004
Yesterday Gavin Theobald improved estimates for a(42), a(43) and a(45). Congratulations !
Due to a break-in into the server, solutions are no longer accepted automatically, please mail them to me at my human address jwr@math.uni.wroc.pl
January 25, 2004
William Marshall announced a(34)=145.
I have done appropriate corrections to DATABASE.TXT.
November 22, 2003 Two days ago Fumitaka Yura got m(18)<=103 and m(19)<=110 ! Congratulations again !
November 19, 2003 Fumitaka Yura got m(17)<=85 !!! Congratulations !!!
If you do not care about it, you should know it can produce a(272)<=3465. If you do not care about that either, you should know that with a(240)<=2874 it can produce a(512)<=9803<9842 and with a little more care even a(512)<=9785.
All results marked with "Theorem 17" are now derived from m(17)<=85.
It also allowed to verify NrootN conjecture up to 30,000 and almost up to 40,000 with a small gap.
November 18, 2003 Ooops again! I had failed to include the right solution to a(25)<=92, for some strange reason I got a(25)<=100 there, but it has been fixed by Al Zimmermann's submission.
Thank you, Al.

November 17, 2003
On November 15 William Marshall announced a(33)=137.
So far noone has submitted any improvements of known estimates. I have added m(28)<=189 that I had missed before.
I can prove a(n)<=n*Sqrt[n] for n<=28,296.
Also I can prove a(n) <= n1.52.
I have produced an explicit example proving a(1024)<=29236 which beats the known fractal pattern.
November 15, 2003
Finally I got a reasonable first version of this website but a lot of work still has to be done. At least you should be able to submit solutions without my intervention. Please click here for more details and updated results. Yesterday I came up with a cute conjecture: a(n)<=n*Sqrt[n]. I think I can prove it for n below 28,000 and above 7,000,000.
I do not think I will do any search in the nearest future. I desperately need your help to do the search - I am sure you can get better results than I can - I have done nothing fancy as far as the search for small n is concerned. I am going to sort out what is known.
November 12, 2003
I got a(1024)<=29236<29525. This is the smallest power of 2 for which I know a(2k)<(3k+1)/2.
Also I got a(78)<=498 which is the smallest known n so that Log[n,2a(n)-1] < Log[2,3].
Jarek Wroblewski jwr@math.uni.wroc.pl