It is my understanding that, when one is describing time complexity with $\mathcal{O}$, $\mathcal{\Theta}$, and $\mathcal{\Omega}$, one must be careful to provide expressions with regards to the size of the input as opposed to the input itself (particularly in the case of numeric algorithms).
For instance, trial division for integer factorization takes up to $\sqrt N$ divisions to factor the integer $N$. However, the size of the input is the number of bits, $w = \lg(N)$. Thus, integer factorization takes $\mathcal{O}\left(2^{w/2}\right)$ time with respect to the size of the input ($w$ bits).
My question: Given the above, would it be considered correct to write in a CS article (relatively informal--on the scale of a blog post) to write that "integer factorization takes $\mathcal{O}(\sqrt{N})$ time with respect to the input variable $N$," and assume that the reader realizes they should substitute $w=\lg(N)$ to obtain the time complexity with respect to the size of the input?