We study the large N limit of some supersymmetric partition functions of the U(N) k × U(N) −k ABJM theory computed by supersymmetric localization. We conjecture an explicit expression, valid to all orders in the large N limit, for the partition function on the U(1) × U(1) invariant squashed sphere in the presence of real masses in terms of an Airy function. Several non-trivial tests of this conjecture are presented. In addition, we derive an explicit compact expression for the topologically twisted index of the ABJM theory valid at fixed k to all orders in the 1/N expansion. We use these results to derive the topologically twisted index and the sphere partition function in the ’t Hooft limit which correspond to genus g type IIA string theory free energies to all orders in the α′ expansion. We discuss the implications of our results for holography and the physics of AdS4 black holes.