We propose an efficient computational method for finding all solutions $n\leq U$ to the Diophantine equation $aσ(n) = bn + c$, where integer coefficient $a,b,c$ and an upper bound $U$ are given. Our method is implemented in SageMath computer algebra system within the framework of recursively enumerated sets and natively benefits from MapReduce parallelization. We used it to discover new solutions to many published equations and close gaps in between the known large solutions, including but not limited to hyperperfect and $f$-perfect numbers, as well as to significantly lift the existence bounds in open questions about quasiperfect and almost-perfect numbers.

翻译：我们提出一种高效的计算方法，用于寻找丢番图方程 $aσ(n) = bn + c$ 的所有解 $n\leq U$，其中给定整数系数 $a,b,c$ 及上界 $U$。我们的方法在 SageMath 计算机代数系统中实现，基于递归可枚举集框架，并原生受益于 MapReduce 并行化。我们利用该方法发现了许多已发表方程的新解，填补了已知大解之间的空白，包括但不限于超完全数与 $f$-完全数，并显著提升了关于拟完全数与几乎完全数的开放问题中的存在性上界。