We study the asymptotic growth of the degeneracy of the BPS local operators
with scaling dimension n/2 in the three-dimensional superconformal field
theories describing N M2-branes. From the large N supersymmetric indices we
obtain the asymptotic formulas for degeneracies of the M2-brane SCFTs according
to the Meinardus theorem. We observe an intriguing universal n2/3 growth
of the degeneracies in various theories of M2-brane SCFTs. We also determine
the coefficients of n2/3 growth as well as further corrections in these
theories explicitly.Comment: 59 page