In one of this question's bonuses I asked you to design a permutation on the natural numbers such that the probability of a random term being odd was \$1\$. Now let's kick it up a notch. I want you to design and implement a permutation, \$f\$, on the natural numbers such that, for every integer \$n\$ greater than 0, the probability on \$f\$ of a member being divisible by \$n\$ is \$1\$.
Definition of Probability
To avoid confusion or ambiguity I am going to clearly lay out what is meant by probability in this question. This is adapted directly from my original question.
Let us say we have a function \$f\$ and a predicate \$P\$. The probability of a number fulfilling \$P\$ will be defined as the limit of the number of members of \$f\{1..n\}\$ fulfilling the predicate \$P\$ divided by \$n\$.
Here it is in a nice Latex formula
$$ \lim_{n\to\infty} \dfrac{\left|\left\{x : x\in \left\{1\dots n\right\}\right\},P(f(x))\right|}{n} $$
In order to have a probability of something being \$1\$ you don't need every member to satisfy the predicate, you just need the ones that don't to get farther apart the bigger they are.
This is code-golf so answers will be scored in bytes with less bytes being better.
You may take the natural numbers to either include or exclude 0. Both are fine.