Consider the following:
These tables show the progression of possible primes removed within the first three primorials, 2, 6 and 30, after sieving by 2, 3 and 5. Numbers are presented in the primorial notation, which I explained here:
Red numbers are those removed in the current round of sieving, orange were previously removed, and black remain as candidate primes.
For the second and third primorials I’ve presented the numbers in two different organizations – the first with numbers written from top to bottom, so that the lower primorial is presented in column format, and the second in traditional left to right enumeration, wrapping down to the next row again at the left.
We see different patterns emerge with the two different formats, which are more apparent when examining the next primorial, 210, shown both ways as follows:
In the first form, with lower primorials in column form, we have rows of non-primes and possible primes, with new non-primes (multiples of the prime currently being sieved) at seemingly random locations.
In the second form, all the new non-primes are in the left-most column (being multiples of 7), and this entire column is now ruled out as non-prime.
I will continue to mull these over to see if anything interesting occurs to me. Meanwhile, one might care to note that the emphasized box in the last table includes all numbers within 7# less than 7^2, so all possible primes in this box are prime.