On the existence of a particular type of finite measure on $\mathbb N$

Question

Let $\mathbb N$ denote the set of all positive integers. Does there exist a countably additive measure $\mu : \mathcal P(\mathbb N) \to [0,\infty)$ such that $\mu(\mathbb N)<\infty$ and $\mu(\{nk: k\in \mathbb N\})=\dfrac 1{n \log n},\forall n>1$ ?