<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] If   $n \ne 2 \!\!\! \mod 3$ and $n \ne 3 \!\!\! \mod 5$: $\;\;$ Write $\;n=3+5+(n-8)$ provided $n \ge 10$.
If $n \ne 2 \!\!\! \mod 3$ and $n = 3 \!\!\! \mod 5$: $\;\;$ Write $\;n=9+5+(n-14)$ provided $n \ge 16$.
If  $n =2 \!\!\! \mod 3$ and $n \ne 3 \!\!\! \mod 5$: $\;\;$ Write $\;n=3+25+(n-28)$ provided $n \ge 30$. $\quad$   Observe that 26=2+5+19.
If $n =2 \!\!\! \mod 3$ and $n = 3 \!\!\! \mod 5$: $\;\;$ Write $\;n=9+25+(n-34)$ provided $n \ge 36$.  $\quad$ Observe that 23=3+7+13.
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