Another refinement is to initially list odd numbers only, , and count in increments of in step 3, thus marking only odd multiples of . This actually appears in the original algorithm. This can be generalized with wheel factorization, forming the initial list only from numbers coprime with the first few primes and not just from odds (i.e., numbers coprime with 2), and counting in the correspondingly adjusted increments so that only such multiples of are generated that are coprime with those small primes, in the first place.
Another refinement is to initially list odd numbers only  and count in increments of in step 3 thus marking only odd multiples of  This actually appears in the original algorithm This can be generalized with wheel factorization forming the initial list only from numbers coprime with the first few primes and not just from odds (ie numbers coprime with 2) and counting in the correspondingly adjusted increments so that only such multiples of are generated that are coprime with those small primes in the first place