Eccoci.

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 $1 \le i,k \le n$.

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

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

Find the sum of all $S(F(k))$ over $3 \le k \le 44$.