  We calculate the superconformal indices of the $\mathcal N=2$ superconformal
field theories realized on $N$ coincident D3-branes in 7-brane backgrounds with
constant axiodilaton via the AdS/CFT correspondence. We include the finite-$N$
corrections as the contribution of D3-branes wrapped around 3-cycles in the
internal space. We take only single-wrapping contributions into account for
simplicity. We also determine the orders of the next-to-leading corrections
which we do not calculate. The orders are relatively high, and we obtain many
trustable terms. We give the results for $N=1,2,3$ explicitly, and find nice
agreement with known results.
