TITLE: An example of a mapping $\eta\colon\mathbb{N}\to\mathbb{N}$ such that $\eta(x)=n$ has infinitely many solutions for each $n\in\mathbb{N}$
QUESTION [8 upvotes]: Suppose I have the mapping $\eta\colon\mathbb{N}\to\mathbb{N}$ such that for each $n\in\mathbb{N}$ the equation $\eta(x)=n$ has infinitely many solutions. I saw this question which is basically the same thing, but some of the examples/solutions are a bit over my head.
I was wondering if someone might know of a closed-form definition for $\eta(x)$ that is relatively simple that accomplishes the goal at hand. There doesn't seem to be anything all that simple even though it is easy to understand the problem. I was thinking of trying to use a trigonometric function, the floor or ceiling function, something modular, etc., but nothing is working out very nicely. Any examples of some mapping definitions that might work out nicely?

REPLY [3 votes]: Here is another simple example:
Write $n$ in binary (or any other basis) and define $\eta$ to be the erasing of the first digit.
This has the desired property because for all integers $x$, as long as $2^m >x$ we have 
$$\eta(2^m+x) =x$$