A362816 - OEIS

A362816

Lexicographically earliest sequence such that nowhere is a term a(n) contained in an arithmetic progression of length greater than a(n).

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

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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

Neal Gersh Tolunsky, Table of n, a(n) for n = 1..10000

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. A362815, A363011 (indices of record highs), A003278, A090822, A281579.

Sequence in context: A140361 A237769 A187182 * A176208 A375422 A330623

Adjacent sequences: A362813 A362814 A362815 * A362817 A362818 A362819

KEYWORD

nonn

AUTHOR

Samuel Harkness, May 04 2023

STATUS

approved