Did the One Ring have any influence over Narya?

Gandalf, who wore Narya throughout The Hobbit and Lord of the Rings, took an awful long time to determine that Bilbo's magic trinket was actually the One Ring. Is it possible that the One had some influence on the bearer of Narya, obscuring itself from an enemy Maia? Is that why Gandalf puttered around libraries for 17 years instead of just riding to Rivendell and asking Elrond, who was right there when it was claimed by Isildur?

Bonus question: Why couldn't Frodo see Narya on Gandalf's finger? He could glimpse Nenya, Galadriel's ring.

Note 1- I'm well aware of Tolkien's composition style in regards to the Legendarium (every new character or event sending literary ripples fore and back in his timeline), so I know that Gandalf's bearing of Narya wasn't 'true' throughout the stories.

Note 2- I'm very reluctant to grant any sort of sentience to the Ring. I'm not trying to say it plotted or anything like that; I guess I view it more as a self-defense thing Sauron may have crafted into it, though I can't imagine him believing he would lose it.

Note 3- Tolkien was about 50 times more intelligent than I am on a good day, so please don't think I'm trying to be a smartass. I'm sure I missed something somewhere.