My students learn this as a "blues scale". I encourage them to memorize the pattern (ie in half-steps or frets, it's 3-2-1-1-3-2) so they can transpose it to any key. They love it! -- so simple and versatile.

Sounds fine with an extra half-step ie A C C# D D# E G (or 3 1 1 1 1 3 2). And now the scale works with both the A major and A minor chords too, because the C# is the major third - give it a go and see what you think! The C natural in the original scale produces a harsh dissonance against the C# in the major chord. So, playing the C# as well resolves that dissonance in a satisfying way for the listener.

And btw, just to be technical, I'd call that flattened third in the C blues scale an Eb, not a D# -- not that it makes any difference in sound of course, but simply because in the scale of C, E is the third note, and D is the 2nd. So if you are raising the second note, it's called a D#. If you are flattening the third, it's an Eb.