I don’t think there is a proper name for these so I will refer to them as “phactors”. Basically, a phactor is a way to sum up to a number using positive real integers that are non zero and not equal to the number. For example, a phactor of 6 would be 1,2,3. It could also be 2,4 or 1,5. There are a total of 10 unique phactors for 6 as per my calculations. My question is if there is a pattern to the total number of phactors of a given number. This was being discussed on a discord group and we computed them up to 15 to try and spot a pattern but could not find any. I tried using finite differences to see if there is some polynomial satisfying it but there is none. Here’s the attachment from the discord conversion. Numbers and their total number of phactors
Has there been any work done on this before? Is there even a proper name other than “phactor” for this? Most importantly, is there a pattern?