FYI, those Germany clusters above can also be swapped around and still mostly work (ie. for the diamond, the CH on the outside with the Hansa in the middle), depending on what fits best for resources/rivers and other adjacencies. But it should be incredibly hard to place a Hansa without at the very least a base +5 bonus, not counting city-states.
You do have to plan how to mazimize factories, since it's not efficient to build them in every city. But especially mid-game, they're a civ that can get new cities up the fastest. Plop a city, place a Hansa, and chop a forest, and the district is basically complete. And then you're probably talking about 12 from the Hansa, plus another 3 from a factory, plus another handful for tiles, and you very well can be pulling in 20 production very early in a city's life without even dedicating a trader to them.
Japan's adjacencies bonuses, though, have the huge edge that they'll boost the hard to get adjacencies. So while in general, I find it hard to get more than +1 or +2 from a Theatre from another civ, Japan can easily surround it with a couple districts and now has a +3 or +4 theatre square. It can also help for Harbors - if you have an inlet tile, you can potentially get 2-3 districts around a Harbor, and if you can get it up to a +4 or +5 bonus, that works really nice with the double harbor adjacency card and a shipyard, although that's still a lot of production needed, plus potentially a card that's not the most valuable to run unless if you have a lot of coastal cities.
But overall, I tend to find a gap between 4-5 as optimal. I don't like "wasted" tiles in the empire (just wait until that one tile in the middle nobody has settled on is your only Niter source, and now you either need to add a city there or wait a hundred turns for your borders to spread there), but obviously if you pack cities too closely, they can easily run out of space, especially if you have lots of water or mountains in the way. But on the flipside, again, not every city needs to be productive. As long as you can get a campus down, the city's a net positive for you. I find the ideal empire has about 4 "productive" core cities that I'll focus on and try to optimize, and the rest are most just fillers that I don't need to focus on.