Consider the divisors of $p_n\# = 30$ for $n = 3$. You can either write them as vectors $(1,0,1)$ meaning $2 \mid x$ and $5 \mid x$ but $3\nmid x$ or simply as a number ($10 = 2\cdot 5$ in this case) which is the "involved divisors of each nonzero entry" multiplied together.
Now the natural path in the following graph starts at $30$ because each of $2,3,5$ divides $0$ (usually we start something at $0$ or $1$, here we chose $0$). Next you add $1$ to $0$ to go to node $1$ since none of $2,3,5$ divide $1$. Then adding one again you get to node $2$ because $2 \mid 2$ but $3,5 \nmid 2$. Then to $3$, same analogous deal, then back to $2$ because $2 \mid 4$ but $3,5 \nmid 4$, and so on... The edges are labeled with the commatized list of steps that traversed them.
I colored steps in the path from $1,\dots, 15$ green while steps $16, \dots 30$ are labeled blue.
Notice the symmetries between blue and green. Green starts at $30$, goes all the way down to $2$, then to $3$, back to $2$, then draws out a square on the left. OTOH, starting at $15$, blue goes all the way up to $1$ (17) then to $6$ (18) back to $1$ (19) and then draws out the same analogous square that green did but this time on the right.
Question.
How do we mathematically describe this symmetry, what is the significance of it, why is the symmetry there, and can it be applied to the question of infinitude of twin primes some how - perhaps in a generalization (infinitude of pairs of numbers $n, n+k$ divisible by certain radicals)?
Note that everything repeats modulo $30$ or modulo $p_n\#$ in general, obviously. So, the same path will be traversed again and again, in the same way.
Link to editable graph, drawn using varkor's Quiver app. Especially, the handling of parallel arrows has nice default behavior in this app.
