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 $$ \prod_{ p \mid n} \left( 1- \frac{ 1}{ p^2} \right) \prod_{ q \nmid n} \left( 1- \frac{ 3}{ q^2} \right) \frac{ n^2}{ 2} + O(n^{ 3/2+o(1)}) $$ 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)
Categories: math.NT math.CO