TITLE: Slicing up $\mathbb{N}\setminus\{1\}$
QUESTION [5 upvotes]: Let $\mathbb{N}$ denote the set of positive integers. For any prime $p\in\mathbb{N}$ let $p\mathbb{N} = \{np: n\in \mathbb{N}\}$. Is there a partition ${\cal P}$ of $\mathbb{N}\setminus\{1\}$ such that for all $B \in {\cal P}$ and every prime $p\in\mathbb{N}$ we have $|B \cap p\mathbb{N}|=1$?

REPLY [5 votes]: Place each $n$ that is not a prime power into its own $B=B(n)$. Then fill the rest of $B(n)$ with powers of primes that do not divide $n$.