Notation for separating out factors of a number

Question

I have an integer (let's call it $n$), and I want to define it as the product of two values: one that's a pure power of two, and another that is odd. Obviously, these two values are unique for a given integer.

How do I write this without sounding stupid? I considered this:

Let $n$ be a number. Define $b$ and $p$ to be the values for which $n = b \times 2^{p}$ is true and $b \equiv 1\ (mod\ 2)$.

There has to be a better way, right? Ideally using commonly known functions or shorthand for factorization.

Answer 1

[Note: of course for one integer $n$, there is no such decomposition.]

The most common way to say this is indeed some variation of "Write $n$ as $2^r m$ with $m$ odd." If you really want alternatives (there might be contexts where these read better due to placing emphasis on $r$ or on $m$), you might also consider:

• "Let $2^r$ be the largest power of $2$ dividing $n$" (this gives you the option of defining this directly if you care more about the factor than the exponent).
• "Suppose $2^r \| n$", where $\|$ is understood as the "prime power which exactly divides" relation.
• "Let $r = v_2(n)$", with $v_2(\cdot)$ being the $2$-adic order function.
• "Let $m$ be the odd part of $n$", abusing the fact that the term "odd part" is reasonably hard to misinterpret when applied to an integer (as opposed to, say, a function $f :\mathbb R \to \mathbb N$).

But for most purposes the first formulation is the go-to idiom, even though it sounds clunky when you (intentionally, I presume) string it out to an extreme length.