A392461 a(n) is the minimum diameter of a n-element set of integers for which no nonzero d has more than 2 representations as a difference of elements of the set.

0, 1, 2, 4, 6, 9, 13, 18, 23, 29, 36, 44, 53, 63, 74, 84, 97, 111

Offset

1,3

Comments

a(n) is the minimum diameter of a 2-Golomb Ruler.

a(n) >= (n^2-2n^{3/2}+n)/2+sqrt(n)-1, by a version of the Erdos-Turan argument for Sidon sets.

a(1) through a(10) computed by Hassenklover (1984).

a(q) <= (q^2-1)/2 - 1, if q is a prime power, proved by Caicedo, Martos and Trujillo (2015).

a(n) ~ n^2/2, proved by Caicedo, Martos and Trujillo (2015).

a(n) >= (n^2 - (2-e)n^{3/2})/2 for some e>0 and sufficiently large n, proved by Balogh, Füredi and Roy (2023).

a(n) >= (n^2 - (2-2^{-12})n^{3/2})/2 for sufficiently large n, proved by Carter, Hunter and O'Bryant (2025).

References

M. D. Atkinson and A. Hassenklover, Sets of integers with distinct differences, Sch Comput. Sci., Carleton Univ., Ottawa, Ont., Canada, Rep. SCS-TR-63, Aug 1984.

M. D. Atkinson, N. Santoro, and J. Urrutia, Integer sets with distinct sums and differences and carrier frequency assignments for nonlinear repeaters, IEEE Transactions on Communications, Vol. Com-34, No. 6, June 1986.

J. Balogh, Z. Füredi and S. Roy, An upper bound on the size of Sidon sets, Amer. Math. Monthly 130 (2023), no. 5, 437-445.

Y. Caicedo, C. Martos, and C. Trujillo, g-Golomb rulers, Rev. Integr. Mat. 33 (2015), no. 2, 161-172.

D. Carter, Z. Hunter and K. O’Bryant, On the diameter of finite Sidon sets, Acta Math. 175 (2025), no. 1, 108-126.

C. Martos, D. Daza and C. Trujillo, Near-optimal g-Golomb rulers, IEEE Access 9 (2021), 65482-65489.

Links

Table of n, a(n) for n=1..18.

Aditya Gupta, C++ optimized multithreaded code for computing 2-golomb rulers.

Aditya Gupta, Optimal Diameters of High Multiplicity g-Golomb Rulers, arXiv:2605.14229 [math.CO], 2026. See p. 2.

Kevin O'Bryant, Code for generating g-Golomb Rulers.

Example

a(6) = 9: The set {0, 1, 3, 5, 8, 9} has differences 1,2,2,3,1,3,4,5,4,5,7,6,8,8,9, with no difference appearing more than 2 times, so a(6)<=9. That a(6)>=9 requires exhaustive search.

Crossrefs

Cf. A003022, A392462, A392463.

Sequence in context: A079717 A247179 A319158 * A175780 A114830 A177239

Adjacent sequences: A392458 A392459 A392460 * A392462 A392463 A392464

Keyword

nonn,more,hard,changed

Author

Kevin O'Bryant, Jan 13 2026

Extensions

a(15)-a(17) from Aditya A Gupta, Feb 24 2026

a(18) from Mikhail Aristarkhov, Jun 09 2026

Status

approved