Prime-Sum Numbers (noch nicht übersetzt)

Define function P(n,k) = 1 if n can be written as the sum of k prime numbers (with repetitions allowed), and P(n,k) = 0 otherwise.

For example, P(10,2) = 1 because 10 can be written as either 3 + 7 or 5 + 5, but P(11,2) = 0 because no two primes can sum to 11.

Let S(n) be the sum of all P(i,k) over 1 ≤ i,k ≤ n.

For example, S(10) = 20, S(100) = 2402, and S(1000) = 248838.

Let F(k) be the kth Fibonacci number (with F(0) = 0 and F(1) = 1).

Find the sum of all S(F(k)) over 3 ≤ k ≤ 44