Can every positive integer $n > 2$ be written as the sum of $k$ prime numbers?

Question

I immediately see a problem with number $6$, if $1$ is considered non-prime. Are there other numbers that are not possible to write as the sum of $k$ prime numbers?

Note: Prime numbers can be added repeatedly.

Answer 1

Let $n$ be even. Then $n=\underbrace{2+2+2+\cdots}_{\frac{n}{2} \text{times}}$

Let $n$ be odd. Then $n=\underbrace{2+2+2+\cdots}_{\frac{n-1}{2}-1 \text{ times}}+3$

Answer 2

If you allow repetition, then you can just use the prime factorization to generate the necessary sums, as in the examples in comments.