Optimal sinusoidal Erathostenes' Sieve

Question

NOTE: I have simplified this post here. Please, consider reading that post instead of this one. Thanks.

Given the series of prime numbers greater than 9, we can organize them in four rows, according to their last digit ($d=1,3,7$ or $9$), and in $k=1,2,3\ldots$ columns corresponding to the $k$-multiple of $10$ we have to add to those four digits in order to obtain a prime number. Therefore, each prime is identified by a point $P(k,d)$.

I illustrate this representation in the following scheme.

For instance, in correspondence of the column $k=15$ ($x$-axis), we find two points in the rows $d=1$ and $d=7$ ($y$-axis), because $15\cdot 10+1=151$ and $15\cdot 10+7=157$ are primes.

Within this reference system, we can introduce the function

$$ f_1(k)=5+4\cos(\frac{\pi}{3}(k-1)), $$

which pass through some of the points representing the primes (green).

Similarly, we can introduce the function

$$ f_2(k)=5+4\cos(\frac{\pi}{3}(k-2)), $$

which pass (blue) through some other primes, with respect to the ones related to $f_1$:

Conversely, the (orange) function

$$ f_3(k)=5+4\cos(\frac{\pi}{3}(k-3)), $$

pass through some primes related to the green function $f_2$ (in correspondence of $d=3$).

However, by means of $6$ functions in the form $f_h(k)=5+4\cos(\frac{\pi}{3}(k-h))$, with $h=1,2,3,4,5,6$ we are able to intercept all the primes:

My question rises from the fact that there is a sort of "multiplicity" of some primes, since they are reached by more than one function. Hence,

Can we reduce the number of these functions, in such a way that each prime is intercepted by one and only one (sinusoidal) function?

Thanks for your comments and suggestions. I apologize in case of naivety/incorrectness.

EDIT: Thanks to the answer of Yves, I realized that the question might be not clear. Therefore, please, see also my own answer for further clarifications.

Answer 1

These sinusoids cover all integers ending in $1,3,7$ or $9$ ($1$ and $9$ once and $3$, $7$ twice). You are investigating the odd integers non-multiple of $5$, not the primes.

This plot of primes is aperiodic and the answer to your question is no.

Answer 2

Note, for example, that the period of $u_4(k)=5+2\cos \dfrac{\pi}{3}(k-4)$ is $6$ and that $u_4(4)=7$. That means that $u_4$ covers the arithmetic sequence $t_n = 6n + 7$. Dirichlet's theorem states that this sequence includes an infinite number of primes. Find another sine or cosine waves that covers the sequence of the form $6n+5$ and you will have included all primes except for $2$ and $3$.