Is there a definition that defines the set of all factors of a natural number?

Question

So let's say that you have the number $n=12$. The factors of $12$ are $1$, $2$, $3$, $4$, $6$, and $12$. I'm wondering if there's some definition that when you plug in a natural number $n$, it will give you the set of all of it's factors. So, if I plugged in $12$ for $n$, I would get the set ${1,2,3,4,6,12}$, in no particular order of course. Is there such a thing? If so, what is it?

Edit: Since $1$ and $n$ are always factors of $n$, I can get as far as ${1,n} \subseteq F $ where $F$ is the set of factors of $n$.

Answer 1

Your question indicates that you do know the definition: the set of factors of the positive integer $n$ contains just the positive integers $m$ that divide $n$. If you'd rather not use the word "divide" in the definition, then it's the integers $m$ for which there is an integer $k$ such that $km=n$.

That definition does use words. Writing it more compactly this way (as in @AlbertoDebernardi 's comment):

$$ d(n)=\{m∈N:m | n\} $$

(where "$|$" is the symbol for "divides") doesn't really make it more "mathematical". Those symbols are all defined with words.

When you talk about "plugging in" an integer and getting the set of factors you may be thinking about a computer program. At http://rosettacode.org/wiki/Factors_of_an_integer you will find programs (in many languages) that accept an $n$ as input and return the list of its factors.

The python example:

factors(45)
[1, 3, 5, 9, 15, 45]

Answer 2

You need to define such a set for $n \in \mathbb{N}-\{0\}$ in the following way:

$$\{x\large|n=\prod_i^mp_i^{a_i} \land x=\prod_i^mp_i^{k_i} \,\, \text{where} \,\, 0\le k_i\le a_i\}$$