Yesterday my wife and I went to a takeaway food shop in suburban Sydney called 이레 (transliterated as Irae outside and Irea on the server's aprons). This word was not on my translator app or Google Translate. I asked my (Korean, university-educated) wife and she had to think for a few moments before she tentatively said 'seven days'. I know 하루 and 이틀, so I asked if it was connected to those, and she said yes.
One of my Korean textbooks gives 하루, 이틀, 사흘 and 나흘 in the context of a medical appointment (instructions for taking medicine), but comments that the last two are usually replaced by 3일 and 4일 (presumably pronounced sam-il and sa-il).
What is the origin of the 하루, 이틀 ... series, and how often are those terms used, and in which contexts? They appear to be related (more or less clearly, in some cases) to the Korean numbers, but the second syllable is inconsistent. Is there a pattern I'm missing? Do I really have to learn the numbers between 3 and 6, 8 and 10, or are they too rarely used?