## Prime-Sum Numbers

### Problem 543

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 `k`th Fibonacci number (with F(0) = 0 and F(1) = 1).

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