The kinetics of the reaction of molecular oxygen with hydroperoxyalkyl radicals have been studied theoretically. These reactions, often referred to as second O-2 addition, or O-2 + QOOH reactions, are believed to be responsible for low-temperature chain branching in hydrocarbon oxidation. The O-2 + propyl system was chosen as a model system. High-level ab initio calculations of the C3H7O2 and C3H7O4 potential energy surfaces are coupled with RRKM master equation methods to compute the temperature and pressure dependence of the rate coefficients. Variable reaction coordinate transition-state theory is used to characterize the barrierless transition states for the O-2 + QOOH addition reactions as well as subsequent C3H6O3 dissociation reactions. A simple kinetic mechanism is developed to illustrate the conditions under which the second O-2 addition increases the number of radicals. The sequential reactions O-2 + QOOH -> OOQOOH -> OH + keto-hydroperoxide -> OH + OH + oxy-radical and the corresponding formally direct (or well skipping) reaction O-2 + QOOH -> OH + OH + oxy-radical increase the total number of radicals. Chain branching through this reaction is maximized in the temperature range 600-900 K for pressures between 0.1 and 10 atm. The results confirm that n-propyl is the smallest alkyl radical to exhibit the low-temperature combustion properties of larger alkyl radicals, but n-butyl is perhaps a truer combustion archetype.