A323419 -id:A323419 - cp's OEIS frontend

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

Tanya Khovanova and PRIMES STEP Junior, Jan 13 2019

nonn

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.

			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.

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

Cf. A005836, A074940, A323418, A323419.

A323418 The third greedy 3-free sequence on nonnegative integers.

Original entry on oeis.org

7, 8, 16, 17, 19, 20, 34, 35, 43, 44, 46, 47, 55, 56, 88, 89, 97, 98, 100, 101, 115, 116, 124, 125, 127, 128, 136, 137, 163, 164, 250, 251, 259, 260, 262, 263, 277, 278, 286, 287, 289, 290, 298, 299, 331, 332, 340, 341, 343, 344, 358, 359, 367, 368, 370, 371, 379, 380, 406, 407, 487, 488, 736, 737, 745, 746, 748, 749

Offset: 1

Tanya Khovanova and PRIMES STEP Junior, Jan 14 2019

nonn

Consider the lexicographically earliest sequence of nonnegative integers 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 next earliest 3-free sequence, which is sequence A323398. We get this sequence on the next iteration.

When represented in ternary this sequence consists of integers ending in 1 or 2, and there is exactly one digit 2 before that that might be followed by zeros.

Cf. A005836, A074940, A323398, A323419.

cp's OEIS Frontend

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

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