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 $3\le k\le 44$.