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  two  segments, which can be visualized as a  two-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  29th  NE  diagonal intersecting with several special factorable functions.
When the intersection coincides with a  lattice point the natural number is composite.
Note: 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.   29 | f(29)  and  29 | f(29 + 29 · k)  or  31 | f(30)  and  31 | f(30 + 31 · k).

The Ulam two-quarter spiral (see first picture) becomes a continuous spiral (see below) when the prime spiral is folded together. Clearly visible is the translation  n   to  n - 1  at the original seam.

The Ulam two-quarter spiral can also be presented in a diamond shape.


Prime spirals: The Ulam two-quarter spiral as prime spiral with two segments.