This is especially true of the Prime shells.  Any power of a prime number will only have itself as a factor, so shells that are the powers of Prime shells should have exactly the same pattern as the Prime shell.

In The 3 Shell, Reveal Shells in black, Prime shells in red, Echoes shells in blue , Reveal shells are black, Prime shells are red, Echoes and Powers of 2 are cyan, all other Echoes are blue, Powers of any number are green, and all other numbers are yellow.  The diagram extends to 37, to show 6^2, ie the echo of the reveal shell of 3.

Prime shells, the red ones, are defined as having no numbers excluded except for the zero line and/or the self line, ie Ɵ=0 or 2π  (0° or 360°).