  We consider two competing sets of nuclear magic numbers, namely the harmonic
oscillator (HO) set (2, 8, 20, 40, 70, 112, 168, 240,...) and the set
corresponding to the proxy-SU(3) scheme, possessing shells 0-2, 2-4, 6-12,
14-26, 28-48, 50-80, 82-124, 126-182, 184-256... The two sets provide 0+ bands
with different deformation and band-head energies. We show that for proton
(neutron) numbers starting from the regions where the quadrupole-quadrupole
interaction, as derived by the HO, becomes weaker than the one obtained in the
proxy-SU(3) scheme, to the regions of HO shell closure, the shape coexistence
phenomenon may emerge. Our analysis suggests that the possibility for
appearance of shape coexistence has to be investigated in the following regions
of proton (neutron) numbers: 8, 18-20, 34-40, 60-70, 96-112, 146-168,
210-240,...
