Your peaks at $A=4n$ are the so-called "alpha-particle nuclei," which have even proton number, even neutron number, and $N=Z$. In a very hand-wavy way you can say that alpha-particle nuclei are especially tightly bound because the alpha particle is especially tightly bound.
Inside of a nucleus you have separate orbitals for neutrons and protons. The magic numbers you're thinking of, where a nucleon orbital closes, do correspond to especially tightly bound nuclei, but to see it you need to look at a full table of isotopes. For instance, tin (with $Z=50$) has ten or eleven stable isotopes, while indium and antimony ($Z=49,51$) have only two stable isotopes each. There are also six different elements with stable isotopes having $N=50$, which kind of stands out on the chart. There are some "doubly-magic" nuclei with closed shells in both $N$ and $Z$, such as calcium-40, calcium-48, which are both stable, and tin-100, tin-132, both near the drip lines.