Forcing among patterns with no block structure

Define the following order among all natural numbers except for 2 and 1: $$4\gg 6\gg 3\gg \cdots \gg 4n\gg 4n+2\gg 2n+1\gg 4n+4\gg \dots$$ Let $f$ be a continuous interval map. We show that if $m\gg s$ and $f$ has a cycle with no division (no block structure) of period $m$ then $f$ has also a cycle with no division (no block structure) of period $s$. We describe possible sets of periods of cycles of $f$ with no division and no block structure.