Problem

Source: Iran 3rd round 2012-Final exam-P8

Tags: number theory proposed, number theory



a) Does there exist an infinite subset $S$ of the natural numbers, such that $S\neq \mathbb{N}$, and such that for each natural number $n\not \in S$, exactly $n$ members of $S$ are coprime with $n$? b) Does there exist an infinite subset $S$ of the natural numbers, such that for each natural number $n\in S$, exactly $n$ members of $S$ are coprime with $n$? Proposed by Morteza Saghafian