Abstract The existence of magic numbers of protons and neutrons in nuclei is essential for understanding nuclear structure and fundamental nuclear forces. Over decades, researchers have conducted theoretical and experimental studies on the new magic number { $Z(N) = 6$}, focusing on observables such as radii, binding energy, electromagnetic transition, and nucleon separation energies. We perform the {\it ab initio} no-core shell model calculations for the occupation numbers of {the lowest} single particle states in {the ground states of} {$Z(N)=6$ and $Z(N)=8$ isotopes (isotones)}. Our calculations do not support {$Z(N)=6$} as a magic number over a span of atomic numbers. However, $^{14}$C and $^{14}$O exhibit the characteristics of double-magic nuclei.