CWMO 2004 Problem 4

Let $\mathbb{N}$ be the set of positive integers. Let $n\in \mathbb{N}$ and let $d(n)$ be the number of divisors of $n$ . Let $\varphi(n)$ be the Euler-totient function (the number of co-prime positive integers with $n$ , smaller than $n$ ).

Find all non-negative integers $c$ such that there exists $n\in\mathbb{N}$ such that $$d(n) + \varphi(n) = n+c ,$$ and for such $c$ find all values of $n$ satisfying the above relationship.