  We consider a family of Argyres-Douglas theories, which are 4D $\mathcal N=2$
strongly coupled superconformal field theories (SCFTs) but share many features
with 4D $\mathcal N=4 $ super-Yang-Mills theories. In particular, the two
central charges of these theories are the same, namely $a=c$. We derive a
simple and illuminating formula for the Schur index of these theories, which
factorizes into the product of a Casimir term and a term referred to as the
Schur partition function. While the former is controlled by the anomaly, the
latter is identified with the vacuum character of the corresponding chiral
algebra and is expected to satisfy the modular linear differential equation.
Our simple expression for the Schur partition function, which can be regarded
as the generalization of MacMahon's generalized sum-of-divisor function, allows
one to numerically compute the series expansions efficiently, and furthermore
find the corresponding modular linear differential equation. In a special case
where the chiral algebra is known, we are able to derive the corresponding
modular linear differential equation using Zhu's recursion relation. We further
study the solutions to the modular linear differential equations and discuss
their modular transformations. As an application, we study the high temperature
limit or the Cardy-like limit of the Schur index using its simple expression
and modular properties, thus shedding light on the 1/4-BPS microstates of
genuine $\mathcal N=2$ SCFTs with $a=c$ and their dual quantum gravity via the
AdS/CFT correspondence.
