This note presents the restricted divisor functions and describes the amicable pairs and their iterations and the iterated stable numbers derived from these new functions. These arithmetic functions are related to (2)-type Mersenne primes 2^p − 1, (2, 3)-type Mersenne primes 2ⁱ · 3^j − 1 and multiperfect numbers. Using Mathematica 11.2, we have tabulated all iterated stable numbers and iterated amicable pairs up to 2²⁰. We establish several interesting arithmetical properties of the iterated restricted divisor functions. The tables obtained suggested many open problems on iterated stable numbers, and iterated amicable pairs from iteration of the restricted divisor functions, some of which are proven here. For any natural number n, we studied the order of its iteration relative to the restricted divisor functions. Moreover, the generating series of the iterated restricted divisor functions are found.

中文翻译：

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迭代稳定数和迭代友好对

本笔记介绍了受限除数函数，并描述了友好对及其迭代以及从这些新函数派生的迭代稳定数。这些算术函数与 (2) 型梅森素数 2 ^p  − 1、(2, 3) 型梅森素数 2 ⁱ  · 3 ^j  − 1 和多重完美数有关。使用 Mathematica 11.2，我们已经列出了所有迭代稳定数和迭代友好对高达 2 ²⁰。我们建立了迭代受限除数函数的几个有趣的算术性质。获得的表格提出了许多关于迭代稳定数的开放问题，并从受限除数函数的迭代中迭代了友好对，其中一些在这里得到了证明。对于任何自然数n，我们研究了它相对于受限除数函数的迭代顺序。此外，还找到了迭代受限除数函数的生成级数。