A382870 Minimum period of an optimum covering of the set of integers by translates of its subset with diameter no greater than n, maximized over such subsets.
1, 2, 4, 5, 8, 8, 13, 13, 27, 27, 45, 53, 66, 109, 129, 147, 147, 170, 192, 250, 286, 317
Offset: 0
Examples
For n = 3, the set S = {0, 1, 3} (diameter 3) covers the set of integers Z when translated by ...0, 1, 5, 6, 10, 11... with primitive period 5. Periodic coverings of Z by translates of S with smaller period are possible (e.g. by taking the entire Z as the set of translations) but they have greater density of overlaps and thus are not optimal. A different set can have a different period of an optimal covering, e.g. {0, 3} has the minimum period of 2 achieved by translations by ...0, 2, 4..., but a(n) maximizes over the subsets of diameter n, and the maximum is attained by S, so a(3) = 5.
Links
- Béla Bollobás, Svante Janson, and Oliver Riordan, On covering by translates of a set, Random Structures and Algorithms, 38 (2011), 33-67; arXiv:0910.3815 [math.CO], 2009-2010. See Table 1.