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A323398 Lexicographically first 3-free sequence on nonnegative integers not containing the Stanley sequence S(0,1), which is A005836. 2
2, 5, 6, 11, 14, 15, 18, 29, 32, 33, 38, 41, 42, 45, 54, 83, 86, 87, 92, 95, 96, 99, 110, 113, 114, 119, 122, 123, 126, 135, 162, 245, 248, 249, 254, 257, 258, 261, 272, 275, 276, 281, 284, 285, 288, 297, 326, 329, 330, 335, 338, 339, 342, 353, 356, 357, 362, 365, 366, 369, 378, 405, 486, 731, 734, 735, 740, 743, 744 (list; graph; refs; listen; history; text; internal format)
OFFSET

1,1

COMMENTS

Consider the lexicographically earliest sequence of nonnegative numbers that does not contain the arithmetic mean of any pair of terms (such sequences are called 3-free sequences as they do not contain 3-term arithmetic progressions): 0,1,3,4 and so on. This sequence is Stanley sequence S(0,1). Remove numbers in the Stanley sequence from nonnegative integers and repeat the process of finding the earliest 3-free sequence. The result is this sequence.

When written in base three these are numbers that contain exactly one 2 that might be followed by zeros.

LINKS

Table of n, a(n) for n=1..69.

Tanya Khovanova and Kevin Wu, Base 3/2 and Greedily Partitioned Sequences, arXiv:2007.09705 [math.NT], 2020.

EXAMPLE

Removing the Stanley sequence from nonnegative integers we get sequence A074940:  2, 5, 6, 7, 8, 11, 14, 15, 16, 17, 18 (Numbers having at least one 2 in their ternary representation). Our new sequence starts with 2,5,6. It can't contain 7 as 5,6,7 form an arithmetic progression. It can't contain 8 as 2,5,8 form an arithmetic progression. The next term is 11.

CROSSREFS

Cf. A005836, A074940, A323418, A323419.

Sequence in context: A002133 A092306 A319242 * A233865 A090552 A024520

Adjacent sequences:  A323395 A323396 A323397 * A323399 A323400 A323401

KEYWORD

nonn

AUTHOR

Tanya Khovanova and PRIMES STEP Junior, Jan 13 2019

STATUS

approved

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Last modified December 4 02:40 EST 2021. Contains 349469 sequences. (Running on oeis4.)