IR-MO 2012 Problem 8

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