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A362816 Lexicographically earliest sequence such that nowhere is a term a(n) contained in an arithmetic progression of length greater than a(n). 3
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, 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, 5, 5, 5, 3, 3, 3, 5, 5, 5, 6, 5, 5, 3, 2, 2, 5, 2, 2, 6 (list; graph; refs; listen; history; text; internal format)
OFFSET
1,1
COMMENTS
Progressions are terms at indices in arithmetic progression and with values which are some arithmetic progression too.
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.
Conjecture: only terms in A362815 appear in this sequence. This is true through the first 10^5 terms.
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.
LINKS
Samuel Harkness, MATLAB program
EXAMPLE
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.
PROG
(MATLAB) See Links section.
CROSSREFS
Cf. A003278, A090822, A281579, A362815, A363011 (indices of record highs).
Sequence in context: A140361 A237769 A187182 * A176208 A330623 A153095
KEYWORD
nonn
AUTHOR
Samuel Harkness, May 04 2023
STATUS
approved

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Last modified April 20 16:19 EDT 2024. Contains 371844 sequences. (Running on oeis4.)