This is a question about a particular sieve, which interests me because it has implications for twin primes. Consider primes of the form $p_i=6k_i \pm 1$. In a sequential manner, starting with $p_1=5$, for each such prime, mark for removal natural numbers $a \equiv \pm k_i \bmod p_i$. My question is: After each round of this sieving procedure, what is the length of the longest stretch of consecutive numbers that have been marked for removal? If a specific value for that length cannot be formulated, can some upper bound be placed on it?
I do not know of a conceptual basis for attacking this problem, but I am hoping some member of the community might. By brute force, I have analyzed the first three rounds. In analyzing round $i$, it is only necessary to analyze stretches among numbers up to $a \le q=\prod p_i$, as any pattern in that range repeats for larger numbers $b=a+nq$.
After round 2 (up to $5 \times 7=35$), the longest such stretch is $4$, which occurs twice beginning at $13$ and $19$. After round 3 (up to $5 \times 7 \times 11=385$), the longest stretch is $6$, which occurs four times beginning at $151,159,221,229$. Unsurprisingly, a stretch that begins at some value $x$ is reflected in a similar stretch that ends at $q-x$. Analyzing the residue classes of $5005$ members of round 4 is beyond my capacity to work by hand.
For each new round involving $p_i$, the numbers marked for removal will be spaced apart alternately by $2k_i$ and $p_i-2k_i \approx 4k_i$. In view of this, the expectation is that any particular stretch from earlier rounds may be lengthened by at most $2$, or two such stretches may be fused if they are separated by no more than one unmarked number.