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

The counterclockwise  Ulam spiral  with  startvalue 59  played an important role in unraveling the  Ulam spiral  mystery.  The  SE main diagonal is interesting, since it is rich with prime numbers.
Refinding the same sequence  (for n > 29)  on the  59th SE diagonal  led to the  59th SE function  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  59th SE 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  59th SE 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.   29 | f(14)  and  29 | f(14 + 29).

The last picture gives an overview of relevant intersections of  special factorable functions  with the  59th SE function.


The Ulam spiral: The  59th SE diagonal   in a Cartesian coordinate system