Construct a function with pre-specified behaviour

Question

Could you suggest a function $f:\mathbb{N}^+\setminus\{1\}\rightarrow \mathbb{N}^+$ such that

1. $\lim_{x\rightarrow \infty}\frac{f(x)}{x}=0$
2. $\lim_{x\rightarrow \infty}f(x)=\infty$
3. $f(x)<x$ $\forall x \in \mathbb{N}^+$
4. $f(500)=340$

where $\mathbb{N}^+$ denotes the strictly positive natural numbers (zero excluded), $\mathbb{N}^+\setminus\{1\}$ denotes $\mathbb{N}^+$ without $1$. See a related question here which imposes less constraints on the desired function. For example, the answer to that question suggests $$ f(x)=340*(\log(x^2+1)/\log(500^2+1)) $$ which does not work here because $f(x)>x$.

Answer 1

Let's say $f$ is in $\Theta(\sqrt x)$, which will satisfy the first three conditions (with the help of a minimum). To satisfy the fourth, we add a scaling factor: $$f=\min\left(x-1,\left\lfloor\frac{340}{\sqrt{500}}\sqrt x\right\rfloor\right)$$