A323398 Lexicographically first 3-free sequence on nonnegative integers not containing the Stanley sequence S(0,1), which is A005836.

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

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: A092306 A349158 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