GCD generator

Let $R$ be a subset of $\mathbb N \times \mathbb N$ defined as follows:

$R= \{ (a,b) \in \mathbb N \times \mathbb N: b$ is the lowest natural number with $\gcd(b+1, 2b+1)=a\}$ .

For each $(a,b)$ , let $A$ denote $(a+b)$ . What is sum of all possible values of $A$ ?

$$ Notations:

• $\mathbb N$ denotes the set of natural numbers .

• $\gcd(\cdot)$ denotes the greatest common divisor function.