"anti-prime" is just Brady trying to rebrand an old boring name to sound cooler (as he does). "highly composite" has a perfectly well defined meaning already, trying to shift it to shoehorn in some artificial distinction between it and "anti-prime" is counterproductive. If the aim is to produce a more descriptive name for a similar sequence of numbers that includes 480, I think GP's "weakly highly composite" is much better. I might propose just "weakly composite" or "mediocrely composite".
Edit: Actually, this was (also) already done by Ramanujan and has a better name than any proposed here: http://oeis.org/A067128 - "largely composite numbers" which does in fact contain 480. Perhaps every sharded system should choose a shard count from this list?
Thanks for correcting me, but I don't think drawing a distinction between "anti-prime" (clearly specific) and "highly composite" (vague) is counterproductive. On the other hand, "highly" and "largely" mean roughly the same thing, so treating "highly composite" and "largely composite" as distinct in this way is linguistically counterintuitive and counterproductive. Likewise, "weakly highly" reads like ambiguous nonsense when you don't go out of your way to specially define it.
Edit: Actually, this was (also) already done by Ramanujan and has a better name than any proposed here: http://oeis.org/A067128 - "largely composite numbers" which does in fact contain 480. Perhaps every sharded system should choose a shard count from this list?