A384729 A B_2-sequence with reciprocal sum > 2.1615.

1, 2, 4, 8, 13, 21, 31, 45, 66, 81, 97, 123, 148, 182, 204, 252, 291, 324, 352, 415, 486, 540, 651, 706, 792, 838, 928, 1046, 1134, 1228, 1358, 1407, 1512, 1624, 1869, 1938, 2087, 2170, 2367, 2480, 2608, 2765, 3033, 3080, 3232, 3567, 3605, 3797, 3950, 4267, 4505, 4677, 5064, 5290, 5480, 5655, 6059, 6507, 6892, 6967

Offset: 1

Author

Logan J. Kleinwaks, Jun 08 2025

Keywords

• nonn

Comments

This is the B_2 sequence with largest reciprocal sum that is known to the author, as of the date of submission. The reciprocal sum of the first 1010 terms, which are given in the attached b-file, is 2.16150003... This already provably exceeds the reciprocal sum of the infinite sequence by R. Lewis (A046185), defined by greedy extension of a sequence of 68 terms (e.g., compute R. Lewis' sequence to 1600 terms, then bound the remainder using the result of B. Lindström on the maximum cardinality of B2-sequences with elements in [1, N]). By extending this sequence greedily after the first 1010 terms, a larger reciprocal sum can obviously be achieved.

This sequence first differs from A046185 at the 25th term.

This sequence was found by the author by using a combination of beam search and batch greedy algorithm, as part of an experiment to evaluate LLM code generation and mathematical reasoning (all code was written by the LLM, though with significant prompting).

Links

• Logan J. Kleinwaks, Table of n, a(n) for n = 1..1010
• Rachel Lewis, Mian-Chowla and B2 sequences, 1999.
• Bernt Lindström, An inequality for B_2-sequences, Journal of Combinatorial Theory, Volume 6, Issue 2 (1969), 211-212.
• Eric W. Weisstein, B_2-Sequence.. From MathWorld--A Wolfram Web Resource.

Crossrefs

Cf. A046185 (R. Lewis), A005282 (Mian-Chowla).