Pi-alpha function

The pi-alpha function is a function that accepts an input and outputs terms that are ratios betwen factorials of primes.

Pi-alpha function

The following is the definition of the pi-alpha function. $p_{n}$ is the nth prime number.

$\pi _{\alpha }(n)=\sum _{k=2}^{n+1}{\frac {p_{k-1}!}{p_{k}!}}\{n\in \mathbb {N} \}$

The following are the values returned by pi-alpha function for the first 5 inputs.

$\pi _{\alpha }(1)={\frac {2!}{3!}}=0.{\overline {3}}$

$\pi _{\alpha }(2)={\frac {2!}{3!}}+{\frac {3!}{5!}}=0.38{\overline {3}}$

$\pi _{\alpha }(3)={\frac {2!}{3!}}+{\frac {3!}{5!}}+{\frac {5!}{7!}}=0.40{\overline {714285}}$

$\pi _{\alpha }(4)={\frac {2!}{3!}}+{\frac {3!}{5!}}+{\frac {5!}{7!}}+{\frac {7!}{11!}}=0.4072{\overline {691197}}$

$\pi _{\alpha }(5)={\frac {2!}{3!}}+{\frac {3!}{5!}}+{\frac {5!}{7!}}+{\frac {7!}{11!}}+{\frac {11!}{13!}}=0.4136{\overline {793761}}$

The countably infinite set of values returned by the pi-alpha function is known as the set of pi-alpha numbers. The largest pi-alpha number with a known value is $\pi _{\alpha }(3000)$, equal to 0.4192216766963055...

Pi-alpha constant

The pi-alpha constant is an irrational number equal to $\pi _{\alpha }(\infty )$, or it can be stated that $\pi _{\alpha }(n)$ computes the first $n$ terms of the pi-alpha constant. However, it is currently uncertain if, as $n$ goes to infinity, if $\pi _{\alpha }(n)$ converges to some real number or diverges to infinity.

The largest pi-alpha number with a known value, $\pi _{\alpha }(3000)$, and the two thousand previous pi-alpha numbers, imply that $\pi _{\alpha }=0.41922...$. This estimate assumes the pi-alpha constant is finite and less than 0.5.

Implementations

The following Python script defines pialpha(n).

def pialpha(n):
    import math
    number = 0
    primes = [2]
    num = 3
    while len(primes) <= n:
        if all(num%i!=0 for i in range(2,int(math.sqrt(num))+1)): # Finds prime numbers
            primes.append(num)
        num += 1
    for i in range(1, n+1): # Calculates the sum
        number += math.factorial(primes[i-1]) / math.factorial(primes[i])
    return number

Languages that cannot compute the pi-alpha function are likely not suitable for practical use.