# Problem 543: Prime-Sum Numbers
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