# What is the pair $(n,k)$ called where $n$ is an integer and $k$ is the ordered factorization index?

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.”

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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.

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