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A072842
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Largest m such that we can partition the set {1,2,...,m} into n subsets with the property that we never have a+b=c for any distinct elements a, b, c in one subset.
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6
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OFFSET
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1,1
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COMMENTS
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The fourth term is at least 66 (Ernst Munter), from { 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 }
Another set of subsets can be described with this sequence of digits (among 8238): 112122213313333333232124144444144422244144441444412223333333331222 (where each digit represents a subset) The fifth term is at least 195 and can be built with the previous sequence, 515, then 66 digits 5 and finally the sequence 122133333333312224144441444222441444444441422213333133331222. I'd like to see a 196-digit sequence. [Julien de Prabere]
Actually a(5)=196 was given by Walker without proof. But Eliahou et al. give an example of such a partition, so a(5) >= 196. And Robilliard et al. give an example for n=6 with [1..574], so a(6) >= 574. - Michel Marcus, Mar 26 2013
To clarify: a(1)-a(4) are known. a(5) = 196 was claimed by Walker but no proof is known, though the value seems likely to be correct. - Charles R Greathouse IV, Jun 13 2013
The best known lower bounds for the next terms: a(6) >= 582, a(7) >= 1740, a(8) >= 5201, a(9) >= 15596. See link to Eliahou's 2017 article. - Dmitry Kamenetsky, Oct 20 2019
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REFERENCES
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EFNet #math, Jul 23 2002 (can we replace this with a link? - N. J. A. Sloane)
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LINKS
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FORMULA
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It is known that 315^((n-1)/5) <= a(n) <= floor(n!*n*e). - Pierre Bornsztein (bornsztein(AT)voila.fr), Sep 02 2003
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EXAMPLE
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a(2) = 8 because we may partition the set {1, 2, ..., 8} into {1, 2, 4, 8} and {3, 5, 6, 7} with the desired property, and this is the unique solution; 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 3+6=9). [Corrected by Julien de Prabere, Dec 17 2009]
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CROSSREFS
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The requirement that a not equal b is the only difference between these numbers and the Schur numbers A045652.
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KEYWORD
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nonn,more,nice,hard
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AUTHOR
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Tor G. J. Myklebust (pi(AT)flyingteapot.bnr.usu.edu), Jul 24 2002
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EXTENSIONS
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More terms from Pierre Bornsztein (bornsztein(AT)voila.fr), Sep 02 2003
Minor additions from Julien de Prabere (jdpbr(AT)aliceadsl.fr), Feb 25 2010
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STATUS
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approved
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