For all positive integers $n, k$, let $f(n, 2 k)$ be the number of ways an $n \times 2 k$ board can be fully covered by $n k$ dominoes of size $2 \times 1$. (For example, $f(2,2)=2$ and $f(3,2)=3$.) Find all positive integers $n$ such that for every positive integer $k$, the number $f(n, 2 k)$ is odd. Please provide the value of f(n, 2 k) when k = 1. The answer should be in the format 2^k-1, directly give the value of m + n.

Let's think step by step and output the final answer within \boxed{}.
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