We introduce the gapped coherent state in the form of a single-photon source (SPS) that consists of uncorrelated photons as a background, except that we demand that no two photons can be closer in time than a time gap [Formula: see text]. While no obvious quantum mechanism is yet identified to produce exactly such a photon stream, a numerical simulation is easily achieved by first generating an uncorrelated (Poissonian) signal and then for each photon in the list, either adding such a time gap or removing all photons that are closer in time from each other than [Formula: see text]. We study the statistical properties of such a hypothetical signal, which exhibits counter-intuitive features. This provides a neat and natural connection between continuous-wave (stationary) and pulsed single-photon sources, with also a bearing on what it means for such sources to be perfect in terms of single-photon emission.