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?
From your description and from what I understand, you seek what is referred to as Positive Integer Partition. The existing literature uses the word "Partition" instead of "phactors". Mathematicians have found beautiful formulas to approximate and to calculate the number of partitions for a positive integer either exactly or approximately. The formulae are not trivial and writing them down here is not useful since there are many references that list them already. The result obtained from the exact formula counts the case $n=n$ as one of the partitions, but in your example you don't. So consider this minor point. The concept is usually discussed in Discrete Mathematics, Number Theory and Combinatorics. Here are some references you may wish to browse first, later you may search for the source that meets the depth you desire.
Theory:
Discrete Mathematics-Google Ebook-Start at page 10.
Stackoverflow: Exact Number of Integer Partitions.
Wiki-Partition Function.
Free Calculators (Web):
Wolfarm Alpha Number Partition Calculator.
Dcode Calculator for Number Partition.