Divisibility-preserving function

Definition

Suppose $f$ is an arithmetic function, i.e., $f$ is a function from the natural numbers to a commutative unital ring $R$. We say that $f$ is divisibility-preserving if, whenever $d$ and $n$ are natural numbers such that $d$ divides $n$, $f(d)$ divides $f(n)$.

Note that whether a function is divisibility-preserving depends on what we consider the target ring.

Relation with other properties

Stronger properties

• Completely multiplicative function