For any positive integer $n$, $\varphi(n)$ is the number of positive integers less than or equal to $n$ which are coprime to $n$. The function $\varphi$ is known as the Euler $\varphi$ function. This function may also be denoted by $\phi$.

Among the most useful properties of $\varphi$ are the facts that it is multiplicative (meaning if $\gcd(a,b)=1$, then $\varphi(ab)=\varphi(a)\varphi(b)$) and that $\varphi(p^{k})=p^{{k-1}}(p-1)$ for any prime $p$ and any positive integer $k$. These two facts combined give a numeric computation of $\varphi$ for all positive integers:

For example,

From equation (1), it is clear that $\varphi(n)\leq n$ for any positive integer $n$ with equality holding exactly when $n=1$. This is because

with equality holding exactly when $n=1$.

Another important fact about the Euler $\varphi$ function is that

where the sum extends over all positive divisors of $n$. Also, by definition, $\varphi(n)$ is the number of units in the ring $\mathbb{Z}/n\mathbb{Z}$ of integers modulo $n$.

The sequence $\{\varphi(n)\}$ appears in the OEIS as sequence A000010.