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For n<=41 the exact values of a(n) are known, see The On-Line Encyclopedia of Integer Sequences.

Rodolfo Niborski verified that

More improvements from Gavin Theobald!

Also he has confirmed that

He conjectures that

More improvements from Gavin Theobald!

Using Gavin Theobald's great results I was able to verify NrootN conjecture for n<=1,408,000 !!!!

Great improvements from Gavin Theobald! Now we know

Now I am very busy, but next week I will see what is the status of NrootN conjecture now.

New improvements from Gavin Theobald! Now we know

New improvements from Gavin Theobald! Congratulations !

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.

New improvements from Gavin Theobald! Congratulations !

Gavin Theobald confirmed a(36)=157. This took about 16 days of processing!

New improvements from Gavin Theobald! Congratulations !

Gavin Theobald again! 8 new improvements! Congratulations !

Gavin Theobald confirmed a(35)=150 by an exhaustive search (5 days of a 1.4 GHz processor work).

Gavin Theobald submitted a lot of new improvements since last update. Congratulations !

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

William Marshall announced a(34)=145.

I have done appropriate corrections to DATABASE.TXT.

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.

Thank you, Al.

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) <= n

I have produced an explicit example proving a(1024)<=29236 which beats the known fractal pattern.

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:

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.

I got a(1024)<=29236<29525. This is the smallest power of 2 for which I know a(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