# A072842

Brendan McKay bdm at cs.anu.edu.au
Sat Nov 2 07:10:26 CET 2002

```As Don pointed out, the example for n=2 is wrong.  The reason why
the value is 8 rather than greater than 8 is also wrong and should
be removed.  Also, the expression "max(m(2))" is silly; m(2) is
just one number, not a set to take a maximum over.

I don't have time, but someone should submit a complete revision
of this sequence.  While you are at it, add the word "distinct"
before "a, b, c" in the definition.  The requirement that a\ne b
is the only difference between these numbers and the Schur numbers
A045652.

For the 4th value, I also find configurations with 66 elements but
can't find anything bigger.  So maybe 66 is the right value but this
should only be put in as a conjecture with Rob's URL citation.

Brendan.

* Rob Pratt <rpratt at email.unc.edu> [021102 02:35]:
> On Fri, 26 Jul 2002, Don Reble wrote:
>
> > > %S A072842 2,8,23,52
> > > %N A072842 Largest m such that we can partition the set {1,2,...,m} into
> > >     n disjoint subsets with the property that we never have a+b=c for
> > >     any a, b, c in any of the subsets.
> > > %C A072842 The fourth number may be erroneous.
> > > %e A072842 max(m(2)) = 8 because we may partition the set into
> > >     {1, 3, 5, 8} and {2, 4, 6, 7} but in no other ways; attempting to
> > >     add 9 to either will produce a set with the property that a+b=c for
> > >     some a,b,c (1+8=9 or 2+7=9)
> > > %A A072842 Tor G. J. Myklebust (pi at flyingteapot.bnr.usu.edu), Jul 24 2002
> >
> >     The example should be { 1 2 4 8 } { 3 5 6 7 }.
> >
> >     a(4) is at least 58.
> > 	{ 1 2 4 7 10 13 16 19 22 25 28 31 34 37 40 43 46 49 52 55 58 }
> > 	{ 3 5 6 12 20 27 41 42 56 57 }
> > 	{ 8 9 11 14 15 18 21 47 48 50 51 53 54 }
> > 	{ 17 23 24 26 29 30 32 33 35 36 38 39 44 45 }
> >     Hmm... If it is exactly 58, my little program would need about 10^15
> >     years to prove it. Don't wait up.
> > --
> > Don Reble       djr at nk.ca
>
> a(4) is at least 66 (due to Ernst Munter):
>
> { 24 26 27 28 29 30 31 32 33 36 37 38 39 41 42 44 45 46 47 48 49 }
> { 9 10 12 13 14 15 17 18 20 54 55 56 57 58 59 60 61 62 }
> { 1 2 4 8 11 16 22 25 40 43 53 66 }
> { 3 5 6 7 19 21 23 34 35 50 51 52 63 64 65 }
>
> See Dr. Dobb's Journal--Solutions to the "Monopoles" Problem
> (http://www.ddj.com/documents/s=896/ddj9912m/9912m.htm).
>
> Rob Pratt
> Department of Operations Research
> The University of North Carolina at Chapel Hill

```