Question 193
Let $n \in \mathbb{N}$ (the set of positive integers), and $d(n)$ be the number of positive divisors of $n$. Next, $\varphi(n)$ denotes the number of integers in the closed interval $[1, n]$ which are co-prime with $n$.
Find all non-negative integers $c$, such that there exists $n \in \mathbb{N}$ satisfying
$$
d(n) + \varphi(n) = n + c.
$$
Find all non-negative integers $c$, such that there exists $n \in \mathbb{N}$ satisfying
$$
d(n) + \varphi(n) = n + c.
$$