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

In the counterclockwise  Ulam spiral  with  startvalue 41,  the  SW main diagonal  is rich with prime numbers.
When using the   startvalue 0  the same sequence  (for n > 20)  is found on the  41th SW diagonal  in a Cartesian coordinate system, as shown above  (prime numbers are in   red,  composite numbers in   grey).

Downwards for smaller natural values of  n  of the  41th SW function  the line spirals inwards,
faster at each crossing of a  | y | = | x |  line,  see  the unraveling of the Ulam spiral.

When the intersection of a  special factorable function  with the  41th SW function  coincides with a  lattice point  
the member is composite, see the examples below.
Note: for a divisor  d  applies that  d | f(n)  and also  d | f(n + d),  thus e.g.   41 | f(20)  and  41 | f(20+ 41),  
with  f(20) = 1681 = 41 · 41  and  f(61) = 15047 = 41 · 367.

The last picture gives an overview of relevant intersections of  special factorable functions  with the  41th SW function.


The Ulam spiral: The  41th SW diagonal   in a Cartesian coordinate system