What is the most general distribution for which the expected value of the multiplicative inverse equals the multiplicative inverse of the expected value?

Motivation: I'm into modelling dynamics on graphs and I found a problem which is easily solvable in cases where the degree distribution of the vertices is a distribution where $E[1/k]=1/E[k]$. (${k}_{i}$ is the degree of the ith vertex) From this solution I may gain an insight into how to unify multiple models.

So particularly I'm looking for a distribution which consists of non-negative, finite integers. But I'm also interested in continuous solutions. Distributions where $E[1/{k}^{n}]=1/E[{k}^{n}]$ may also help unifying the models.

What I do know so far, that ${k}_{i}=1$ is a particular solution. In the continuous case every function where $f(x)=f(1/x)$ and $E[x]=1$ is a solution. I know what momentum generating functions are and they seem like a good direction to try in, but I failed so far.

What is the most general form of this distribution? Does it have a name? It sounds like something trivial, like a "famous" distribution, but I can't find it.