Sieving in a Nutshell

Sieving is a very misunderstood subject. Eventually I intend to expand on the below, which explains some of the principles involved. This is V0.0 presently!

Basically, the maths says that the density of primes in a large set of related numbers is dependent on which small primes can be divisors of members of that set. In particular, most of the time each prime p contributes a thinning of the prime density in the set by a (p-c)/p factor for each prime which can be a divisor, and c is the number of residues mod p that will generate p as a factor. If p cannot be a divisor of any member in the set, then the factor still holds, it's just that c=0.

The final density is the product of all these factors.

Arithmetic Progressions

Ordinary ranges of numbers and arithmetic progressions, which include sets such as fixed-n Proth-form families, have _every_ prime in this product, and c=1 for every prime.

Numbers like the above (arithmetic progressions) with a primorial scale factor only have primes greater than the prime in the primorial in their product. Again c=1 for those primes, and c=0 for the primes in the primorial component. e.g., numbers of the form: