Kempner function

In number theory, the Kempner function ${\displaystyle S(n)}$ is defined for a given positive integer ${\displaystyle n}$ to be the smallest number ${\displaystyle s}$ such that ${\displaystyle n}$ divides the factorial ${\displaystyle s!}$. For example, the number ${\displaystyle 8}$ does not divide ${\displaystyle 1!}$, ${\displaystyle 2!}$, or ${\displaystyle 3!}$, but does divide ${\displaystyle 4!}$, so ${\displaystyle S(8)=4}$.

This function has the property that it has a highly inconsistent growth rate: it grows linearly on the prime numbers but only grows sublogarithmically at the factorial numbers.