I’m developing a number class (as in Object-Oriented Programming) and am wondering what to call it. At its core, it represents an integer, but in a way in which not all integers are unique. What it does treat as unique is the pair $(n,k)$, where $n$ is the represented integer and $k$ is the “flavor” (factorization) of that integer. Here, for example, are the 13 unique factorizations of the number 30:
n#k
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30#1 = 2 x 3 x 5
30#2 = 2 x 5 x 3
30#3 = 2 x 15
30#4 = 3 x 2 x 5
30#5 = 3 x 5 x 2
30#6 = 3 x 10
30#7 = 5 x 2 x 3
30#8 = 5 x 3 x 2
30#9 = 5 x 6
30#10 = 6 x 5
30#11 = 10 x 3
30#12 = 15 x 2
30#13 = 30
Wolfram MathWorld defines these as ordered factorizations: “An ordered factorization is a factorization (not necessarily into prime factors) in which $a$ × $b$ is considered distinct from $b$ × $a$.”
Thus, in the notation n#k, $k$ is an index into the list of ordered factorizations of $n$. I have methods to map from the index $k$ to its respective factorization, and vice-versa.
What I’m looking for is a word (or very short phrase) to go before the word “number” or “integer,” e.g., “indexed number” or “indexed integer.”
Well, I’m going to go with “factorized number” for this. It’s not so much the index that’s important to me as it is the fact that there is a number and and index. Together they imply a particular (ordered) factorization of a number.
At first I was thinking “factored number,” but it seems that “factoring” a number means to decompose it into the product two factors rather than the product of a list of two or more factors, which would imply that a factored number is a number that has simply been split into the product of some pair of its factors.
On the other hand, “factorization” consistently means the decomposition of a number into the product of two or more factors, e.g., prime factorization or prime-power factorization, which would imply that factorized number is a number that has been decomposed into the product of some subset of its factors.