This is a question a student asked me when I was teaching a class. When I use the notation $\sum_{d|n}f(d)$, each divisor is only counted once without multiplicity. If $n=4$, then $\sum_{d|4}f(d)=f(1)+f(2)+f(4)$ instead of $\sum_{d|4}f(d)=f(1)+2f(2)+f(4)$. But instead of just telling the student, this is just what this notation is defined. It is the convention. I can also say, oh you can think of $f(4)$ as "counting the multiplicity" but I am also not satisfied with this answer as well. Are there any more insightful explanations from experts in number theory?
For example, in the definition of the divisor sum functions $\sigma_k(n)$, we don't count the multiplicity of a divisor. But is there a reason that "counting the multiplicity of a divisor" not so interesting? Or there are people who seriously studied it and I just missed an important part of literature.
As per comments below, maybe let me try to define "multiplicity" here, which is by no means standard definitions in number theory, there are two possible ways I can think of (when that student asked this question, they didn't think very hard about it. They just believe $2$ shouldn't be counted just once as a divisor of $4$):
Definition 1: If $d\ne 1$ is a divisor of $n$, then the multiplicity of $d$ in $n$ is the largest $k$ such that $d^k|n$. The multiplicity of $1$ in $n$ is considered as $1$.
For example, the multiplicity of $2$ in $8$ is $3$. The multiplicity of $6$ in $12$ under this definition is $1$.
Definition 2: If If $d\ne 1$ is a divisor of $n$, suppose $d=p_1^{r_1}\cdots p_l^{r_l}$ and $n=d=p_1^{s_1}\cdots p_k^{s_k}$ with $l\le k, r_i\le s_i$, then the multiplicity of $d$ in $n$ is $\binom{s_1}{r_1}\cdots \binom{s_l}{r_l}$.
For example, the multiplicity of $6$ in $12$ under this definition is $2$, since $6=2\times 3$ and there are two choices for $2$ in the prime decomposition of $12$.
Consider that the “divides” or “is-a-factor-of” relation is a partial order (reflexive, antisymmetric, and transitive). So in a sense, a sum $\sum_{d|n} f(d)$ is analogous to $\sum_{d=0}^n f(d)$, in that it sums over all integers “less than” $n$ in the poset $({\bf N}, |)$.