Recent surveys confirm early results about a deficiency or even absence of CN-strong stars on the asymptotic giant branch (AGB) of globular clusters (GCs), although with quite large cluster-to-cluster variations. In general, this is at odds with the distribution of CN band strengths among first ascent red giant branch (RGB) stars. Norris et al. proposed that the lack of CN-strong stars in some clusters is a consequence of a smaller mass of these stars that cannot evolve through the full AGB phase. In this short paper we found that the relative frequency of AGB stars can change by a factor of two between different clusters. We also find a very good correlation between the minimum mass of stars along the horizontal branch (Gratton et al. 2010, A&A, 517, A81) and the relative frequency of AGB stars, with a further dependence on metallicity. We conclude that indeed the stars with the smallest mass on the HB cannot evolve through the full AGB phase, being AGB-manqué. These stars likely had large He and N content, and large O-depletion. We then argue that there should not be AGB stars with extreme O depletion, and few of them with a moderate one.