Let $f:\mathbb{N} \to \mathbb{C}$ be given by
$$\sum_{d\mid n} f(d) = \phi(n)$$
where $\phi(n)$ is the Euler function, and for all $n \in \mathbb{N}$.
Find all such $f$.
Any hints are welcome. If you wish to post an answer, please only post hints, especially if they are conceptual / fundamental hints regarding elementary number theory or abstract algebra.
My work:
I know that the Euler function $\phi(n)$ counts the number of positive integers up to and including $n$ that is coprime to $n$, that is, $\gcd(k,n) = 1$ for $1\le k \le n$.
So I know that $\phi(1) =1$, $\phi(2)=1$, $\phi(3)=2$, $\phi(4)=2$, $\phi(5) = 4$, $\phi(6) =3$, $\phi(7)=6$ ...
And I observe that, for $n$ prime, I have the pattern:
$$\phi(n) = (n-1)$$
Also, the Euler function is multiplicative so that $\phi(ab) = \phi(a)\phi(b)$
whenever $a$ and $b$ are coprime.
And $$\sum_{d|m} \phi(d) = m$$
How could I proceed from here?
Thanks,