"a long time" or just "long time"

I would always employ this phrase "a long time" in sentences so as to refer to "a period of time." For instance, I would say, "I've worked on this project for a long time," or "A long time ago, there once lived a witch on the hilltop." The other day I came across a sentece in which the phrase was used with the omission of the indefinite article "a." Substituting this shorter phrase for the original ones in the above two sentences, I find it awkward-sounding, but I don't know why. Is it grammatically acceptable? Or to say the least of it, is it frequently used by most intermediate learners or speakers of English?

Thanks!