Do I Need a P-Trap?

What about it?

The toilet above this has a slight sewer odor, but it's only noticeable when the lid is down for a while and is then opened. This doesn't happen with the other toilets in the house, but they're upstairs and I can't see under them. This toilet sits in top of the flange. It's not caulked at all, and I don't see signs of seepage. Any advice is appreciated!

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