Is there a correlation between the apparent color of the sky and the color of the Sun?
The color of the atmosphere is complementary to the color the sun appears, with respect to the total set of frequencies reaching the observer's color sensor.
That is, some frequencies of the light from the sun will be scattered by the atmosphere and the atmosphere will appear to be that color. The sun will appear to be a blend of the colors that are not scattered out.
The atmosphere could also have particles acting as a filter, which absorb certain frequencies of light (or, high enough up, reflect them back into space) causing both the atmosphere and the sun to appear more one color and less of another. For example, a certain kind of dust might cause the sky and the sun to appear a lot more reddish than they otherwise would, akin to wearing rose-colored glasses but at a larger scale.
Third, you can play with the frequencies emitted by the source. The above paragraphs assume that the sun emits at a high level across the whole visible spectrum, but this need not always be true of other suns.
Fourth, and interacting with the third, you can change what frequencies of electromagnetic radiation are visible to the inhabitants of your world. Even this Earth looks quite different to non-fictional creatures that can see better in the UV range.
Finally, you could fictionally change the way people perceive colors. I assume that when you say things like "purple" and "green" above you are referring to particular wavelengths of light that you and I have both been taught are associated with those labels, but there's no good way to know (yet) if your experience of "purple" is the same as mine; the mental experience of "purple" for you might be very similar to my experience of "green" and vice versa.