Fabricate Frequently Factored Functions

Question

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

$$ \lim_{n\to\infty} \dfrac{\left|\left\{x : x\in \left\{1\dots n\right\}\right\},P(f(x))\right|}{n} $$

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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.

Answer 1

Husk, 11 bytes

!uΣz:NCNmΠN

Try it online! Or showing the stats for the first 100 n in the first 5000 terms

Explanation

        mΠ     Map factorial over 
          N    the Natural numbers
      CN       Cut into lists of lengths corresponding to the natural numbers
   z:N         ZipWith append the natural numbers and ^
  Σ            Concatenate into one list
 u             Remove duplicates
!              Index

Proof (Without the u)

For triangle numbers : f(½n*(n+1))=n

Otherwise : f(n)=k! for some k.

As the density of triangular numbers is zero, this has the same probability as the sequence of factorials: 1