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A140516
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For definition see comments lines.
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0
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1, 3, 6, 8, 9, 13, 16, 17, 19, 20, 21, 26, 31, 32, 36, 42, 43, 45, 49, 50, 52, 53, 54, 58, 62, 65, 66, 67, 71, 78, 80, 82, 85, 87, 89, 90, 97, 98, 103, 106, 108, 112, 113, 114, 116, 120, 122, 123, 124, 128, 129, 134, 135, 139, 141, 143, 145, 147, 148, 153, 155, 157, 161
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OFFSET
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1,2
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COMMENTS
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Inspired by Van der Waerden's Theorem, the positive integers are partitioned into two disjoint sequences, A and B, so that they obey the following rules:
Rule 1: For a given n, let A(n) and B(n) be the subsets of A and B with terms less than or equal to n. Any maximal arithmetic subsequence of A(n) must have at least as many terms as any maximal arithmetic subsequence of B(n).
Rule 2: Construct the sequences by starting at 1 and always favoring to add terms to B, unless it violates Rule 1. Add terms to A if they cannot be added to B.
The sequence listed here is the A sequence.
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LINKS
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EXAMPLE
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A gets 1, because B cannot yet have an AS of length 1.
B gets 2, because we favor adding to B and A already has an AS of length 1.
A gets 3, because B cannot yet form an AS of length 2.
B gets 4, because A now has an AS of length 2.
B gets 5, because 2,4,5 does not contain an AS of length 3.
A gets 6, because 2,4,5,6 would contain an AS of length 3.
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MATHEMATICA
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CheckMaxASFromEnd[A_, x_, m_] := Module[{i, n=Length[A], d, k}, If[ m>n+1, Return[False]; , ]; If[ And[ m<=n+1, n < 2], Return[True]; , ]; For[ i=n, i>0, i--, d = x-A[[i]]; bot = x-(m-1)*d; If[ bot <= 0, Break[]; , ]; For[ k=x-2*d, k>=x-(m-1)*d, k -= d, If[ MemberQ[A, k], If[ k == x-(m-1)*d, Return[True]; , ]; , Break[]; ]; ]; ]; False ];
ASRace[n_] := Module[{i, m=0, A={}, B={}}, For[i=1, i <= n, i++, If[ CheckMaxASFromEnd[B, i, m+1], If[ CheckMaxASFromEnd[A, i, m+1], m++, ]; A = Append[A, i]; , B = Append[B, i]; ]; ]; {A, B} ];
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CROSSREFS
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KEYWORD
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nonn
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AUTHOR
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STATUS
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approved
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