Is this form useful for to detecting prime numbers, or can it be discarded?
$f(x)= |\sin\frac{\pi x}{P_1} |+|\sin\frac{\pi x}{P_2} |+⋯+|\sin\frac{\pi x}{P_n} | $
Here is an example: $P_1=2; P_2=3$
$f(x)=|\sin\frac{\pi x}{2} |+|\sin\frac{\pi x}{3}|$
The result is two new prime numbers, $5 $ and $7$ for $ x ∈ N$ without inflection point.
With these numbers, the formula can be expanded to obtain the next results for $n^2$.
The result is new prime numbers: $11,13,17,19,23,29,31,37,41,43$ for $x ∈ N$ without turning point.
This series can be continued infinitely to calculate prime numbers.
This is another description of the sieve of Eratosthenes.
Appendix:
With this formula, you can also factorise all the numbers $P ∈ N$ .
$\sin\frac{\pi P}{x} = 0$ for $x∈N$
Example for $P =12$
