Matter radii of the $^{34-40}$Mg nuclei are investigated by self-consistent Hartree-Fock-Bogolyubov calculations assuming the axial symmetry. With the semi-realistic M3Y-P6 interaction, the $N$-dependence of the matter radii observed in the experiments is reproduced excellently. Both the pairing and the deformation play significant roles in an intertwined manner. The $^{35}$Mg nucleus has a smaller radius than the neighboring even-$N$ nuclei, which is attributed to its smaller deformation. In contrast, a neutron halo is obtained in $^{37}$Mg. We point out that, in contrast to the pairing anti-halo effect that may operate on the even-$N$ nuclei, the pair correlation enhances halos in odd-$N$ nuclei, owing to the new mechanism which we call textit{unpaired-particle haloing}. The halo in $^{37}$Mg is predicted to have peanut shape in its intrinsic state, reflecting $p$-wave contribution, as in $^{40}$Mg. The $N$-dependence of the deformation is significant again, by which the single-particle level dominated by the $p$-wave component comes down.
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