<s>[INST] Conjecture: Every $n \geq 20 \in \mathbb{N}$ can be written as a sum of three integers $(\geq 2)$ that are pairwise coprime This question on the sum of pairwise prime numbers piqued my interest, and I started looking at what numbers can be written as the sum of three pairwise coprime numbers (excluding $1$):
$$
\begin{align*}
10&=5+3+2\\
12&=7+3+2\\
14&=7+5+2\\
15&=7+5+3\\
16&=9+5+2\\
18&=11+4+3\\
19&=11+5+3\\
20&=11+7+2\\
21&=11+7+3\\
22&=11+7+4\\
&\ldots
\end{align*}
$$
It's fairly clear why $10$ is the smallest number possible, but I couldn't find any pattern until $20$ where it seems like all the rest follow suit.
It seems after $20$, there are so many possible combinations that every following number can be written as a sum of three integers $(\geq 2)$ which are pairwise coprime. Is this a known result, and, can it be (dis)proved?
EDIT:
Thanks for the responses all, I believe I have devised a nice case proof (inspired by Yuval Peres' case answer!)
Proof
Case 1: $n$ is even
$$
\begin{align*}
n&=6k=2+3+(6k−5)\\
n&=6k+2=4+3+(6k−7)\\
n&=6k+4=2+3+(6k−5)
\end{align*}
$$
Case 2: $n$ is odd
$$
\begin{align*}
n&=12k+1=3+(6k−7)+(6k+5) \text{ with } k≥2\\
n&=12k+3=9+(6k−5)+(6k−1) \text{ with } k≥2\\
n&=12k+5=3+(6k−5)+(6k+7) \text{ with } k≥2\\
n&=12k+7=3+(6k−1)+(6k+5)\\
n&=12k+9=3+(6k−1)+(6k+7)\\
n&=12k+11=3+(6k+1)+(6k+7) \hspace{35pt}\blacksquare
\end{align*}
$$
 [/INST] Assume that $n$ is not the power of sum prime, i.e $n$ cannot be written as $n=p^{\alpha}$ for some prime $p$ and exponent $\alpha$. In this case, we can find two distinct primes $p,q$ that both divide $n$. Clearly $p+q<pq<n$ and so $n-(p+q)$ is a positive integer. This means that we can write
$$n=p+q+(n-(p+q)).$$
The first two terms, $p$ and $q$, are clearly coprime since $p$ and $q$ are prime, and hence all there is left to check is that $(n-(p+q))$ is not a multiple of $p$ or $q$. This follows from the fact that
$$n-(p+q)\equiv -q \neq 0\mod(p)$$
$$n-(p+q)\equiv -p \neq 0\mod(q)$$
since $n$ is a multiple of $p$ and $q$, and both $p$ and $q$ are prime thus are nonzero modulo each other.
In the general case you should be able to do something similar by shifting, since prime powers are 'rare enough' in a certain sense.
</s>