Compositions with 3 Pairwise Coprime Parts

How many ways can we write $n$ as a sum of $3$ positive integers, no pair of which share a common factor? We express this quantity in terms of the number of solutions to a certain class of linear Diophantine equations. This allows us to show that there are $$\prod _{p\mid n}(1-\frac{1}{p^2})\prod _{q\nmid n}(1-\frac{3}{q^2})\frac{n^2}{2}+O(n^{3\mathrm{/}2+o(1)})$$ such compositions, where the products are over primes that respectively do and don’t divide $n$. This strengthens the previous result of Bubbolini, Luca, and Spiga (arXiv:1202.1670)