I'm working on a proof problem that uses $\phi(n)$ and $\tau(n),$ and I am wondering if $\phi(n)$ and $\tau(n)$ are the factors of $n$/relatively prime numbers smaller than $n$, or the number of factors of $n$/relatively prime numbers smaller than $n$.
I just want some clarification so I don't mess up and use the wrong definition in my proof problem.
TL;DR: I'm basically asking if $\phi(n)$ and $\tau(n)$ are a list of numbers, or the # of numbers in that list.
On
$\phi(n)$, known as Euler's totient function or phi function given the number of positive integers not exceeding $n$ and relatively prime to $n$. For example, $\phi(6)=2$ because $1$ and $5$ are the only positive integers not exceeding $6$ that have no factor common with $6$ other than the obvious common factor $1$.
More generally, if $$n=p_{1}^{k_{1}} p_{2}^{k_{2}} \cdots p_{r}^{k_{r}}$$ where $p_{1}, p_{2}, \ldots, p_{r}$ are the distinct primes dividing $n$, then $$ \varphi(n)=p_{1}^{k_{1}-1}\left(p_{1}-1\right) p_{2}^{k_{2}-1}\left(p_{2}-1\right) \cdots p_{r}^{k_{r}-1}\left(p_{r}-1\right) . $$
$\tau(n)$, on the other hand, gives the number of factors of $n$. For example, $\tau(6)=4$ because the only positive divisors of $6$ are $1,2,3,$ and $6$.
$\phi(n)$ is the number of positive integers less or equal to $n$ that are relatively prime to $n$. For example, $\phi(5)=4$ since $\{1,2,3,4\}$ are all relatively prime to $5$. The function $\tau(n)$ is the number of positive divisors of $n$. For example, $\tau(5)=2$ since $5$ has $2$ divisors $\{1,5\}$.