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Let a "sum" be a set {x,y,z} of distinct natural numbers such that x+y=z and let N_m={1,2,...m}. a(n) is the smallest s such that there is no partition of N_s into n sum-free parts.
3

%I #20 Aug 27 2023 04:39:59

%S 3,9,24,67

%N Let a "sum" be a set {x,y,z} of distinct natural numbers such that x+y=z and let N_m={1,2,...m}. a(n) is the smallest s such that there is no partition of N_s into n sum-free parts.

%C a(5) >= 190 (see Blanchard et al. at p. 7). - _Michel Marcus_, Mar 26 2013

%C a(5) >= 197, a(6) >= 583, a(7) >= 1741, a(8) >= 5202, a(9) >= 15597 (see Ahmed et al. at p. 3). - _Stefano Spezia_, Aug 25 2023

%H T. Ahmed, L. Boza, M. P. Revuelta, and M. I. Sanz, <a href="https://doi.org/10.1007/s11139-023-00760-y">Exact values and lower bounds on the n-color weak Schur numbers for n=2,3</a>. Ramanujan J (2023). See Table 2 at p. 3.

%H P. Blanchard, F. Harary, and R. Reis, <a href="http://www.emis.de/journals/INTEGERS/papers/g7/g7.Abstract.html">Partitions into sum-free sets</a>, Integers: electronic journal of combinatorial number theory, 6. 2006.

%e For n=1, a(1)=3 as there is no partition of N_3={1,2,3} into 1-sum-free parts. In the same way a(2)=9...

%Y Cf. A030126, A072842.

%K nonn,hard,more

%O 1,1

%A R. Reis (rvr(AT)ncc.up.pt), May 22 2006