B_2-Sequence

An infinite sequence of positive integers

(1)

also called a Sidon sequence, such that all pairwise sums

(2)

for  are distinct (Guy 1994). An example is 1, 2, 4, 8, 13, 21, 31, 45, 66, 81, 97, 123, 148, 182, 204, 252, 290, 361, ... (OEIS A005282). Halberstam and Roth (1983) contains an accessible account of most known results up to around 1965. Recent advances have been made by Cilleruelo, Jia, Kolountzakis, Lindstrom, and Ruzsa.

Zhang (1993, 1994) showed that

(3)

which has been increased to  by R. Lewis using the non- sequence 1, 2, 4, 8, 13, 21, 31, 45, 66, 81, 97, 123, 148, 182, 204, 252, 291, 324, ... (OEIS A046185). The definition can be extended to -sequences (Guy 1994).

REFERENCES:

Finch, S. R. "Erdős' Reciprocal Sum Constants." §2.20 in Mathematical Constants. Cambridge, England: Cambridge University Press, pp. 163-166, 2003.

Guy, R. K. "Packing Sums of Pairs," "Three-Subsets with Distinct Sums," "-Sequences," and "-Sequences Formed by the Greedy Algorithm." §C9, C11, E28, and E32 in Unsolved Problems in Number Theory, 2nd ed. New York: Springer-Verlag, pp. 115-118, 121-123, 228-229, and 232-233, 1994.

Halberstam, H. and Roth, K. Sequences, rev. ed. New York: Springer-Verlag, 1983.

Mian, A. M. and Chowla, S. D. "On the -Sequences of Sidon." Proc. Nat. Acad. Sci. India A14, 3-4, 1944.

Sloane, N. J. A. Sequences A005282/M1094 and A046185 in "The On-Line Encyclopedia of Integer Sequences."

Zhang, Z. X. "A B2-Sequence with Larger Reciprocal Sum." Math. Comput. 60, 835-839, 1993.

Zhang, Z. X. "Finding Finite B2-Sequences with Larger ." Math. Comput. 63, 403-414, 1994.