오래 is an adverb which means "for a long time", and the others are based on it.
오래간만: 오래 + 간(period, 間) + 만(since) => a long time (since last time). It is only used for the past sense.
오랜만: a contraction of 오래간만. "오래만" or "오랫만" is a wrong spelling.
오랫동안: 오래 + 동안(a period of time) => for a long while. May be used for any period of time, past or future.
오래 and 오랫동안 are adverbs, while 오래간만 and 오랜만 are nouns (add 에 to make them adverbs - 오랜만에).
- 오랫동안 일기를 쓰지 않았어요. (않에요 is wrong. 않아요 is the present tense, and 않았어요 the past)
- 일기를 오랫동안 [오래] 쓰지 않았어요.
- 오랜만이에요 / 오래간만이에요: It's been a long time (since we met last time) - a common greeting.
- 오랫동안 만나지 못해요: I can't see (you / someone) for a long time - Ok but slightly unnatural sentence.
- 오래만 만나지 못해요: Wrong. 오랜만 is only used for the past.
1) 이 음식은 참 오랜(간)만에 먹어 본다: I'm having this dish for the first time in a long while.
2) 그 사람한테선 오랫동안 소식이 없어: I haven't heard from him in a long time.
3) 할머니, 오래 사세요: Have a long life, grandma.
As in 1), 오랜만 and 오래간만 are only for when something has happened again after a long hiatus.
They are not interchangeable with 오래 or 오랫동안. Compare these
- 나는 오랫동안 숙제를 했다: I did my homework taking a long time.
- 나는 오랜만에 숙제를 했다: I've done my homework for the first time in a while. (It's been a while since I did it last time).
오래 is the most common word and can mean anything lasting long: 오래 가다 (last long), 오래 살다(live long), 오래 끌다(drag something a long time), 오래 되다(has been in existence a long time; old, aged), 오래 기다리다(wait a long time), etc.