I don't have much background in analysis or Abstract algebra, but that's why I'm here. I'm looking for an interpolation of y=2-Adic valuation of x (that being in this case that is the elementary defined function where for all integers it outputs the exponent of the greatest power of 2 dividing x) that satisfies this property over all integers, such that the function is continuous ply defined, and differentiable over the real numbers. It is also possible that no such function exists, but I assume there are many such function considering the conditions that had to be added for a unique interpolation of the factorial, namely the gamma function.
Ideally I could actually get such a function, or a proof or understanding of why it doesn't exist, and ideally this function would be defined in the most elementary way possible, perhaps through some sort of series or integral, or whatever methods may work.
Also, for what it is worth, I am aware there is an extension of the p-adic valuation of a number to the rationals, but it need not satisfy those, conditions, just the 2-adic valuation of the integers.
I thank you for whatever time you will take to consider this problem, not matter how small.
It says at http://oeis.org/A007814 that
$$v_{2}(n) = \sum_{r\ge1} \frac{r}{2^{r+1}} \sum_{k=0}^{2^{r+1}-1} e^{\frac{2k\pi i(n+2^r)}{2^{r+1}}}$$