具有3个成对互质部分的合成物
Compositions with 3 Pairwise Coprime Parts
摘要：
我们有多少种方法可以将$n$写成$3$正整数的和，其中没有一对共享公共因子？我们用某类线性丢番图方程的解的个数来表示这个量。这使我们能够表明，有$$ \prod_{p \mid n} \left( 1- \frac{1}{p^2} \right) \prod_{q \nmid n} \left( 1- \frac{3}{q^2} \right) \frac{n^2}{2} + O(n^{3/2+o(1)}) $$这样的组合物，其中的产品是超过素数，分别做和不除$n$。这加强了之前Bubbolini，Luca和Spiga的结果（arXiv：1202.1670）
英文摘要：
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)
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
