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%I A362816 #34 Sep 28 2024 07:39:13
%S A362816 2,2,3,2,2,3,3,3,5,2,2,3,2,2,3,3,3,5,3,5,5,5,3,3,3,5,5,2,2,3,2,2,5,5,
%T A362816 3,3,2,2,3,2,2,5,3,3,5,3,5,5,3,3,5,5,3,5,5,5,6,5,3,5,5,6,5,3,3,3,5,3,
%U A362816 5,5,5,3,3,3,5,5,5,6,5,5,3,2,2,5,2,2,6
%N A362816 Lexicographically earliest sequence such that nowhere is a term a(n) contained in an arithmetic progression of length greater than a(n).
%C A362816 Progressions are terms at indices in arithmetic progression and with values which are some arithmetic progression too.
%C A362816 1 is never in the sequence, because if a(n) = 1, then {a(n),a(n+1)} would form an arithmetic progression greater than 1 in length.
%C A362816 Conjecture: only terms in A362815 appear in this sequence. This is true through the first 10^5 terms.
%C A362816 If this is true, then a(A003278) = 2, because the only way to constrain 2 would be {2,2,2}, and A003278 is defined by adding the smallest term which avoids any 3 term arithmetic progressions. If the conjecture is false, arithmetic progressions {4,3,2}, {8,5,2}, etc. may further constrain 2s.
%H A362816 Neal Gersh Tolunsky, <a href="/A362816/b362816.txt">Table of n, a(n) for n = 1..10000</a>
%H A362816 Samuel Harkness, <a href="/A362816/a362816.m.txt">MATLAB program</a>
%e A362816 For n=9 first we check 1 (never in the sequence). If a(9) were 2, {a(1),a(5),a(9)} = {2,2,2} would form an arithmetic progression of length 3 with a minimum value of 2; this is not allowed. Next, if a(9) were 3, {a(6),a(7),a(8),a(9)} = {3,3,3,3} would form an arithmetic progression of length 4 with a minimum value of 3; this is not allowed. Next, if a(9) were 4, {a(5),a(7),a(9)} = {2,3,4} would form an arithmetic progression of length 3 with a minimum value of 2; this is not allowed. Last, a(9) = 5 fits the definition, as no arithmetic progressions p can be made such that length(p) > min (p) and 5 is the least positive integer where this is satisfied, so a(9) = 5.
%o A362816 (MATLAB) See Links section.
%Y A362816 Cf. A362815, A363011 (indices of record highs), A003278, A090822, A281579.
%K A362816 nonn
%O A362816 1,1
%A A362816 _Samuel Harkness_, May 04 2023

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