Can the Function $p(t, r) = e^{r \sin \left( \frac{(2t+1) \pi}{2} \right)}$ Be Used to derive a Prime Generating Function?

Question

I am exploring a mathematical function related to prime numbers and have written a Python script to test it. Here is the function and the code I've used:

Math Functions:

1. Sum of Divisors Function:

$$ \sigma(n) = \sum_{i=1}^{n} i, \quad \text{where } i \text{ is a divisor of } n.$$

2. Function $ r(x) $:

$$ r(x) = \sigma(x) - x,$$

where $\sigma(x) $ is the sum of divisors of $ x$.

3. Function $p(t, r)$:

$$ p(t, r) = e^{r \sin \left( \frac{(2t+1) \pi}{2} \right)}$$

Python Code:

import math
import cmath

def sum_of_divisors(n):
    # Calculate the sum of divisors σ(n)
    return sum(i for i in range(1, n + 1) if n % i == 0)

def r_function(x):
    # Calculate r(x) = σ(x) - x
    return sum_of_divisors(x) - x

def calculate_p(t, r):
    # Calculate p(t, r) using the given exponent formula
    exponent = r * cmath.sin((2 * t + 1) * math.pi / 2)
    return cmath.exp(exponent)

# Generate r values and calculate p(t, r)
x_values = range(1, 100)  # Adjust range as needed
results = []

for x in x_values:
    r = r_function(x)
    t = 1  # You can adjust this or loop through different t values
    p = abs(calculate_p(t, r))
    results.append((x, r, p))

# Print results
for x, r, p in results:
    print(f"x = {x}, r = σ({x}) - {x} = {r}, p = {p:.6f}")

Observations:

• Whenever $x$ is a prime number, $r(x) = 1$, and consequently, $p(t, r) \approx 0.367879$ (which is close to $e^{-1}$.
• For non-prime $x$, $r(x) > 1$, and the value of $p(t, r)$ tends to $0$ rapidly.

Question:

Can the function $p(t, r)$ be formally used to identify prime numbers? Specifically:

1. Is there a better way to use $r$, or a closed form equation instead of $r(x) = \sigma(x) - x$? The better way should atleast achieve one of: a. Be closed form. b. Gives a prime generating function directly.

2. Can we generate other significant sequences other than the prime numbers sequence by choosing $r$, which ones?

Edit 1: $$p(x) = e^{-\frac{\sin\left(\frac{(x \pi)}{\log(x)}\right)}{\pi}}.$$ Is amazingly intereting!