The (double) Primorial sieve  &  Prime spirals E-mail

The segmented prime spirals are an offspring of the Ulam spiral.
The counterclockwise Ulam spiral with startvalue  0  is a  four-quarter spiral,  and thus a prime spiral with four segments. Downwards is the prime spiral with  one  segment, which can be visualized as a  one-quarter  Ulam spiral.

The picture below shows the special factorable functions that contain no prime numbers > p2.
When a function intersects with a special factorable function, the natural number on the intersection is a composite number.

Clearly visible is the  41th  SE  diagonal intersecting with several special factorable functions.
When the intersection coincides with a  lattice point, the natural number is composite.
For a divisor  d  applies that  d | f(n)  and also  d | f(n + d · k)  with  k = {0, 1, 2, 3, ...}
Thus e.g.   41 | f(41)  and  41 | f(41 + 41 · k)  or for   43 | f(42)  and  43 | f(42 + 43 · k).

Prime spirals: The Ulam one-quarter spiral as prime spiral with one segment.