Classification of particle numbers with unique Heitmann-Radin minimizer

We show that minimizers of the Heitmann-Radin energy (R. C. Heitmann, C. Radin, J. Stat. Phys. 22, 281-287, 1980) are unique if and only if the particle number N belongs to an infinite sequence whose first thirty-five elements are 1, 2, 3, 4, 5, 7, 8, 10, 12, 14, 16, 19, 21, 24, 27, 30, 33, 37, 40, 44, 48, 52, 56, 61, 65, 70, 75, 80, 85, 91, 96, 102, 108, 114, 120 (see the paper for a closed-form description of this sequence). The proof relies on the discrete differential geometry techniques introduced in (L. De Luca, G. Friesecke, preprint).