Robo-AO Kepler Planetary Candidate Survey III: Adaptive Optics Imaging of 1629 Kepler Exoplanet Candidate Host Stars

The Robo-AO \textit{Kepler} Planetary Candidate Survey is observing every \textit{Kepler} planet candidate host star with laser adaptive optics imaging to search for blended nearby stars, which may be physically associated companions and/or responsible for transit false positives. We present in this paper the results of our search for stars nearby 1629 \textit{Kepler} planet candidate hosts. With survey sensitivity to objects as close as $\sim$0.15" and magnitude differences $\Delta$m$\le$6, we find 223 stars in the vicinity of 206 target KOIs; 209 of these nearby stars have not previously been imaged in high resolution. We measure an overall nearby-star probability for \textit{Kepler} planet candidates of 12.6\%$\pm$0.9\% at separations between 0.15" and 4.0". Particularly interesting KOI systems are discussed, including 23 stars with detected companions which host rocky, habitable zone candidates, and five new candidate planet-hosting quadruple star systems. We explore the broad correlations between planetary systems and stellar binarity using the combined dataset of Baranec et al. (2016) and this paper. Our previous 2$\sigma$ result of a low binary fraction of KOIs hosting close-in giant planets is less apparent in this larger dataset. We also find a significant correlation between binary fraction and KOI number, suggesting possible variation between early and late \textit{Kepler} data releases.