A number $n \in \mathbb{N}$ is called quasi-perfect if $\sigma(n) = 2n+1$, where $\sigma$ is the sum of divisors function. It is known that if $n$ is quasi-perfect, then it must be the square of an odd integer (an oddly beautiful Putnam problem from the 1970s). To date no number is known to be quasi-perfect.

My question concerns the existence of 'generalized' quasi-perfect numbers, or rather, let $a,b \in \mathbb{N} \cup \{0\}$ be fixed integers, and call an integer $n$ $(a,b)$ quasi-perfect if it satisfies $\sigma(n) = an+b$.

Are there any known values of $a,b$ for which the number of $(a,b)$ quasi-perfect numbers are known to be infinite?

dblueshad some nice preliminary observations in the very last post of this AoPS thread: artofproblemsolving.com/Forum/viewtopic.php?f=57&t=83696 – Daniel m3 Sep 6 '13 at 16:42